Be grateful for whoever comes, because each has been sent as a guide from beyond. Rumi

Uploaded by
Rumi

119 downloads 4200 Views 2MB Size

on graded rings with finiteness conditions

The happiest people don't have the best of everything, they just make the best of everything. Anony

jordan homomorphisms of rings

Pretending to not be afraid is as good as actually not being afraid. David Letterman

Snap Rings

Live as if you were to die tomorrow. Learn as if you were to live forever. Mahatma Gandhi

Polynomial Rings

Be like the sun for grace and mercy. Be like the night to cover others' faults. Be like running water

on the zeros of polynomials over division rings

Pretending to not be afraid is as good as actually not being afraid. David Letterman

Retaining Rings

At the end of your life, you will never regret not having passed one more test, not winning one more

Reversible skew Laurent polynomial rings, rings of invariants and related rings Nongkhran Sasom

You have survived, EVERY SINGLE bad day so far. Anonymous

Pacific Journal of Mathematics

ON SUBRINGS OF RINGS WITH INVOLUTION P JEK -H WEE L EE

Vol. 60, No. 2

October 1975

PACIFIC JOURNAL OF MATHEMATICS Vol. 60, No. 2, 1975

ON SUBRINGS OF RINGS WITH INVOLUTION PJEK-HWEE L E E

We give a systematic account on the relationship between a ring R with involution and its subrings S and K, which are generated by all its symmetric elements or skew elements respectively.

I. Introduction. Let R be a ring with involution * and 5 the subring generated by the set S of all symmetric elements in R. The relationship between R and S has been studied by various authors. In [3] Dieudonne showed that if R is a division ring of characteristic not 2, then either S = R or SQZ(R), the center of R. Later Herstein [4] extended this result by proving S = R for any simple ring R with dim z i?>4 and char.i?^2. The restriction on characteristic was removed by Montgomery [12]. Recently, Lanski [9] proved that if_R is prime or semi-prime, so is 5. In §2 of this paper, we show that S can inherit a number of ring-theoretic properties such as primitivity, semisimplicity, absence of nonzero nil ideals etc.. In doing so, a notion called symmetric subring, which is a generalization of S and its *homomorphic images, is introduced so that a group of theorems of the same type, including Lanski's results, can be proved via a more or less unified argument. We show also that numerous radicals of S are merely the contractions from those of R. As a consequence, we see that R modulo its prime radical behaves much like S in many respects. In §3 we establish a corresponding theory for K, the subring generated by all skew elements. The only result hitherto known concerning K was as follows [4], [12]: If R is_simple and dimzi? >4, then K - RL As a matter of fact, the subring K2 is more closely related to JR than K is. We apply thejtechnique developed in §2 to study the relationship between R and K2, and then derive some parallel theorems for K. II. Symmetric subrings. Our work depends heavily on the notion of Lie ideals. By a Lie ideal U of R we mean an additive subgroup which is invariant under all inner derivations of JR. That is, [uy x] = ux - xu E U for all u E U and x E R. The following lemma concerning Lie ideals will be referred to frequently in the sequel, and it is a combination of some results in [5]. LEMMA 1. Let Rbe a semi-prime ring and U a subring and Lie ideal of R. Then U contains the ideal of R which is generated by [U,U]. If U is commutative, then u2G Z for all u E U. 131

132

PJEK-HWEE LEE

Rings with involution abound with examples of Lie ideals. One can easily show that any subring, generated by symmetric elements and containing T = {x + x*|JC G R} the set of all traces, must be a Lie ideal. In particular, both S and T are Lie ideals. Another essential property of S follows from the next lemma. We denote by N the set of all norms, i.e. N = {xx*| JC G R}. LEMMA

andxUx*C

2. Let U be an additive subgroup of R such that T C U C S Ufor all x G R. IfN C 17, thenxUx*Q U for all x G R.

Proof. We prove by induction that xux unx * G Ό for all x G R and Mi, , un G (7. The case n = 1 is clear. Assume the assertion holds for n - 1 then XM!W2

UnX * = | > , Mj] [U2

Mn, X * ] + ( X M ^ *>M2

Un + UX(XU2

MnX * )

- u1xx*u2- - - un E U because C7 is a Lie ideal. A subring U of R is called a symmetric subring if: U is generated by a set of symmetric elements. TUNCU xUx*QU for all x <Ξ R.

DEFINITION.

1. 2. 3.

In light of Lemma 2, we know that S is a symmetric subring. From now on, U will always denote a symmetric subring of R. We call an ideal I of R a *-ideal if /* = / . LEMMA 3. // R is semi-prime and I is a *-ideal of R such that I Π U = 0, ί/ien 7 = 0.

Proof. For any a G /, α 2 =

Then / is nil of

Recall that a ring JR is called a *-simple ring if JR2 φ 0 and R has no *-ideal other than 0 and JR. It is well-known that JR is *-simple if and only if either R is simple or JR = A 0 A * for some simple ring A [8, p. 14]. Let Z + = Z Π S. Then if JR is *-simple, we have Z + = 0 or Z + is a field. THEOREM 4. contained in Z + .

// JR is *-simple, then either U = R or U is a field

Proof If U is not commutative, by Lemma 1 it contains a nonzero *-ideal of R so U = JR. Assume that [£/, t/] = 0; then U C S. In this

ON SUBRINGS OF RINGS WITH INVOLUTION

133

case, we need only to prove [/CZ, for if u E U and u^ 0 then If R = A φ A * for some simple ring A, then T = U = S. Thus [C/, I/] = 0 implies [A, A ] = 0 and so R is commutative. If 1? is simple, then U, being a commutative subring and Lie ideal of R, must be central unless 2R = 0 and dimzjR = 4 [5, Theorem 1.5]. So let us examine all possible 4-dimensional cases. If R is a division ring, then x~ιUx = x~ι{xUx*)x = Ux*x CU for all x E i? with x ^ 0. Hence U C Z by the Brauer-Cartan-Hua theorem [7, Theorem 7.13.1,-Cor.]. There remains the case R = F2 where F is a field with char.F = 2. We claim that * must be of symplectic type. Assume the contrary,

[c

[ab

d\

d

for some a E F with ά = α, where - denotes the * induced automorphism on F. Thus

For any α E F, we have Γ0

Lo

l

o J L o oj U o

so a = a. Next, if | , [ab Γ ft [a + c

+d 01Γ0

a + aΛ\a

cj

£ t/ then

0 1 Γ α b U O 0 1 ΓO O l Γ α 6 1 b\ [ab c J L l Oj [ l OJ [ab c_Γ

and hence α = c. But if I a,

1 £ I/, then

Γα 01 Γl O l Γ α bUl 01 LO OJ L0 O j [ a b a \ [ θ OJ yields α = 0 . S o l / = T = | • u• cause T is not a subring. T

,

U

^ f t G F ί which is ridiculous be-

^ \ \a Consequently,

ucs-{[°

U

bT \d , =

bλ

, and b

] c

a M

134

For

PJEK-HWEE LEE

\a

any

bλ

L

\a'

b'Λ^TJ

,

;

.

ε l / , we have

\a

bλ\a'

,

b Ί ^

.

, E r [/τ and

1

hence be = b'c by comparing the diagonal entries of the product. If there exists \a',

b

U with b'/O, then

,]e

ι

where a - c'b'~ .

However, O 01

ΓO 01 Γα'

' Oj Ll o J L c '

bΊTO

;

a

forces ί? = 0, a contradiction. Hence U C (| ILc other hand, if ΓO

cl

ΓO l Ί Γ α

[O OJ L θ O J L c

01

α ' J L l Q\

\\α,c E F\. 1 αj J

0 1 Γ 0 11

αJLo O J

On the

e U

implies c = 0. Therefore, U CZ. Following [11], we say R is *-prime if the product of any two nonzero *-ideals is still not zero. It is easy to see that J? is *-ρrime if and only if aRb = a *Rb = 0 implies a = 0 or b = 0. As a consequence, any + nonzero element in Z is regular in a *-prime ring R. We remind the reader of of a well-known fact that a nonzero Lie ideal of a semi-prime ring always contains elements with nonzero square. THEOREM

5.

If R is *-prime, so is U.

Proof. If [U,U]τ^ 0, then U contains a nonzero *-ideal / of R. For any two *-ideals A, B of U with AB = 0, we have IAIB C AB = 0, so either IAI = 0 or B = 0, ending up with A = 0 or B = 0. Assume that u y o while [U,ί/] = 0. By Lemma 1, there exists u0SU such that UQ E Z but Mo ^ 0. So consider the ring Q of fractions a/a with α E R and α E Z Π t/, α ^ 0. Q is also *-prime with respect to the involution given by (a/a)* = a*/α, and [/' = {u/α E Q) u E [/} is a symmetric subring of Q. As a matter of fact, Q is *-simple. For if J is any nonzero *-ideal of O, / Γl [/V 0 and hence (v/βf^O for some v/β EJΠU'. Since v2EZ, v/β is invertible and so / - Q. By the

ON SUBRINGS OF RINGS WITH INVOLUTION

135

+

previous theorem, U' CZ (Q) and hence U is an integral domain contained in Z+(R). Let CR(V) = {x G R \xυ = vx for all vEV}be the centralizer of a set V in R. LEMMA 6. Let j y 0 be an ideal (or *-ideal) of a prime (resp. *-prime) ring JR. Then CR(I) C Z

/. For a E I, b E CR (I) and JC E i?, we have abx = feαx = αxfe, or equivalently, α(bx - xb) = 0. That is, /[C R (/),R] = 0. Hence [C R (/),K] = 0 and so CR(I)CZ. COROLLARY. Let R be a prime (or *-prime) ring and I a nonzero ideal (resp. *-ideal) ofR such that [I, I] = 0. Then R is commutative. THEOREM

7.

If R is semi-prime, then

Z(U)QZ(R).

Proof Assume first that R is *-prime. If [U, U] = 0, then Z(U) = UCZ(R) by Theorem 5. If [£/, f/]^0, then U contains a nonzero *-ideal / of R, so Z( U) C C* (I) C Z(l?) in view of Lemma 6. In either case, [Z(U),R] = 0. Now assume that R is semi-prime; then R is a subdirect sum of *-prime rings πa(R). Since πa(U) is a symmetric subring of πa(R% we know [ττa(Z(U)\ πa(R)] C [Z(τrα(17)), π β (Λ)] = 0 for all a. Hence, [Z( [/), R ] = 0. The same reduction to *-prime rings together with Theorem 5 gives an alternate proof for Lanski's theorem: THEOREM

8.

If R is semi-prime, so is U.

With this established, we are able to consider the relationship between the prime radicals ?β(R) and ^(U). THEOREM 9.

$ ( 17) = 17 n W

)•

Proof Since U/[UΠ^(R)] - [17 + S$(R)]/ψ(R) which is a symmetric subring of the semi-prime ring JR/^3(jR), so U/[U Π?β(R)] is semi-prime by Theorem 8 and hence ^(U)CUΠ %$(R). On the other hand, if a E U Γ) ^5(1?), then a E U and any m -system in JR containing a must contain 0. [7, Theorem 8.2.3]. Certainly, any m -system in U containing a contains 0. That is, a E^(U). It is well-known that a ring without nonzero nil ideals is a subdirect sum of rings with the following property [6, p. 53]: There exists a nonnilpotent element a such that an(I) E I for all nonzero ideal I

136

PJEK-HWEE LEE

One can impose this condition only on the *-ideals and show that it is a hereditary property. Then, making use of subdirect sum decomposition, we can prove that U inherits the freedom from nonzero nil ideals. Instead of doing this way, we prefer to present a direct proof by considering the nil radical $l(U) of U. THEOREM

10. // R has no nil ideal other than 0, neither does U.

Proof. Let I be the ideal of JR which is generated by [ 17,17]. Since JR possesses no nonzero nil ideal, neither does J, considered as a ring. Hence 9l(U) Π / = 0. For any a G %l(U) and u G [/, we have [a, u] E 9l(U) Π / - 0. Thus 9l(U) C Z(U). Since U is semi-prime by Theorem 8, As an immediate consequence, we have THEOREM 11.

31 (U) = U Π 9l(R ).

Proceed as above with "locally nilpotent" in place of "nil" and with Levitzki radical S in place of 9t, we get THEOREM 12. If R has no nonzero locally nilpotent ideal, neither does 17. THEOREM 13.

£([/) = U Π £ ( # ) .

In [2] the notion of *-primitive ring was introduced as a ring admitting a *-faithful irreducible module M (i.e. Mr = Mr* = 0 implies r = 0). One can easily verify that a ring is *-primitive if and only if it is either primitive or a subdirect sum of a primitive ring and its opposite with the exchange involution. We know that a nonzero ideal of a primitive ring is itself primitive. The proof is applicable to the following more general fact. LEMMA 14. Let R be a primitive (or *-primitive) ring. Suppose that I is a nonzero ideal (resp. *-ideal) of JR, and A is a subring (resp. *-subring, i.e. A* = A) containing I. Then A is also primitive (resp. ^-primitive). THEOREM 15.

If R is primitive or *-primitive, so is U.

Proof If [ 17, U] 7^ 0, U contains a nonzero *-ideal of JR, so it is primitive or *-ρrimitive by Lemma 14. Assume that U is commutative. Then 17 C Z + and every element in JR is quadratic over

ON SUBRINGS OF RINGS WITH INVOLUTION

137

+

Z . Hence R satisfies a polynomial identity. According to Kaplansky's theorem [6, Theorem 6.3.1], R is *-simple and hence U is a field by Theorem 4. Using the fact that a semi-simple ring is a subdirect sum of *-primitive rings, we get immediately THEOREM

16. If R is semi-simple, so is U.

In fact, the semi-simplicity of 5 was first proved by Herstein. His elegant proof was the inspiration of our next theorem which relates the Jacobson radicals of R and U. THEOREM

17. %(U)=UΠ

%(R).

Proof. For a E $(L/) and x E JR, we have ax oax* = ax + ax* + axax* = a(x + x* +

xax*)e%(U)UC%(U).

Thus aR is quasi-regular and hence a GU n $(!?). Conversely, if α ε i / Π S C R ) , α ° & = 0 for some 6 e Λ, then & = 6°(α°fc)* = (f)ofe*)oα*G I/. That is, UΠ^(R) is a quasi-regular ideal of U, so With Theorem 17 in hand, we are ready to study some non-semisimple rings. Following [7], we say R is semi-primary, primary, or completely primary according as R/%(R) is an artinian, simple artinian, or division ring respectively. Since U/$(U) is isomorphic to a symmetric subring of R/J(R)? by Theorem 4 we have THEOREM

18. // R is primary or completely primary, so is U.

As to semi-primary rings, we need some information about the descending chain condition. In a paper [10] which is to appear, Lanski proved that if JR is artinian and \ E R, then so is S. For our purpose, we prove LEMMA

19. If R is semi-prime artinian, so is U.

Proof. By the Wedderburn-Artin theorem, we may write R = -RiΘ'" * ®Rn where each Rt is *-simple. Denote by e, the identity of jRi, then ex E Z + and so eJJex is a symmetric subring of Rx for each /. By Theorem 4, each eJJe, is artinian, so is U = eJJex 0 φ enUen. THEOREM

20. // JR is semi-primary, so is U.

138

PJEK-HWEE LEE

We remark that the assertion corresponding to Lemma 19 for ascending chain condition is not true even if R is a commutative integral domain. A counter example can be found in [13]. Let 9ϊ stand for any of the four radicals Sβ, 2, 9? and $. We have shown 9t(I7) = 17 n 91(1?). If 9t(ϊ7) = U, then C/C9»(JR), so 0 is a symmetric subring of the semi-prime ring R/ΐR(R), and hence ?R(R) = R by Lemma 3. That is, if U is locally nilpotent, nil or quasi-regular, so is R. On the other hand, 91(17) = 0 need not imply 91(1?) = 0. For example, let 1? = F + A be the algebra obtained by adjunction of an identity to a trivial algebra A over a field F with char. Fφ 2. Define (a + a)* - a - a for a E F and a E A. Then S - F is a field, while 9t(JR) = A is a nilpotent ideal. In case A has infinite dimension, this example shows also that JR is not artinian although S is. However, we still have some results on 9Ϊ(JR). For if 91(17) = 0, then the *-ideal 9ΐ(l?) has trivial intersection with 17, hence is nil of index 2. Thus we have aRa = 0 for any a E 9ΐ(l?) and consequently 91(1?) = Besides, U is isomorphic to a symmetric subring of Realizing this fact, one might not be surprised to see that ), instead of 1? itself, satisfies the same properties as U does. 21. Let R be a semi-prime ring and e the identity of Then e is also the identity of JR.

LEMMA

U.

Proof By Theorem 7, eEZ(U)CZ(R). Since e E S, 1 = {x - ex \x E JR} is a *-ideal of 1?. If a - ea E 17, then a — eα = β(α - eα) = 0. Thus / Π [/ = 0 and so / = 0. In other words, e is the identity of JR. The case when JR is semi-prime and S is simple was thoroughly studied by Lanski [9]. An example was given there that 1? is an integral domain but not simple while S is. In the presence of an identity, we have THEOREM 22. Let R be a semi-prime ring. If U is a *-simple ring with identity, so is R.

Proof Let I be any nonzero *-ideal of 1?. Then / Π U^ 0, and the ^simplicity of U implies UCl By Lemna 21, U contains the identity of JR, SO / = 1?. Even if U is a field, 1? can be semi-prime but not simple. The simplest example is the direct sum of two copies of a field with the exchange involution. This example illustrates why we deal with only *-primeness and *-primitivity in what follows.

ON SUBRINGS OF RINGS WITH INVOLUTION

139

THEOREM 23. (1) If U is semi-prime, ψ(R) is nil of index 2. (2) // U is *-prime, so is R/^(R)

Proof We have proved (1) in the discussion before Lemma 21. As to (2), we may assume without loss of generality that R is semiprime. Let / and / be *-ideals of R such that I/ = 0. Then (/ Π U)(J Π U) = 0, so / n U = 0 or / Π U = 0, ending up with / = 0 or J = 0. Suppose that JR is a *-prime ring and I a nonzero *-ideal of R. If / possesses a *-faithful irreducible module M, write M = ml for some m E M and m ^ 0, and define a map from M x R into M by sending (ma, r) to m(ar). One can easily check that such a map is well defined and that M becomes a *-faithful irreducible R -module. This is the content of 24. Let R be a *-prime ring and I a nonzero ideal of R. If I is ""-primitive, so is R. LEMMA

THEOREM 25. (1) // U is semi-simple, then %(R) = ^(R) is nil of index 2. (2) // U is *-primitive, so is R/^(R)

Proof We have seen the proof of (1) earlier. As to (2), we assume that R is semi-prime. By Theorem 23, JR is *-prime. If [U, U]^0, then U contains a nonzero *-ideal / of JR. Lemma 14 shows that I is itself *-primitive and hence JR is also *-primitive by the previous lemma.* If U is commutative, it is *-simple with identity. It follows from Theorem 22 that R is ""-primitive. THEOREM

26. // U is semi-primary, so is R.

Proof It suffices to show that if R is semi-prime and U is artinian, then JR is also artinian. In this case, we have U - UXQ) 0 Un, where each Ui is *-simple artinian. Let e, be the identity of L/i; then eieZ(U)CZ(R). Since 1 = ex+ + en, R=R1φ'"®Rn, with Ri = eR. Each Rt is then semi-prime and contains C7, as a symmetric subring. By Theorem 22 Rt is *-simple, so either Lζ = Ri or L/j is a field. If Ui is a field, then R< satisfies a polynomial identity and hence is a finite dimensional algebra over a field contained in Z(JRr). In either case, Ri is always artinian. Hence R must be also artinian. III. Subrings generated by skew elements. In contrast to S, K is not necessarily a Lie ideal of i?. For instance, in F2 with

140

PJEK-HWEE LEE

char. F^2 and transpose as *, ^ = if "b

A \a,beFγ Although

[Λ ] ro L-i

- 1 1 r o l i p on r i o i r o η o J L - i oJLo o j Lo o J t - i o j

falls outside of K. However, both {x - x * I x E JR}, are always Lie ideals.

K2

and

where

Kl,

Ko =

DEFINITION. By. a skew subgroup V of R we mean a subgroup of R such that K0Q VQK and xVx* C V for all x E JR. Henceforth we shall use V to stand for a skew subgroup of R without further explanation. LEMMA 27.

Proof

V2 is a Lie ideal of JR.

For vu v2 E V and x E R, we have 2

[VιV2, X] = ϋi(lλ>* + X * ϋ 2 ) - ( ^ i X * + Xϋi)ϋ 2 E V .

If Wi,

, wn E V2 and x E R, then [>!

wn,x] = W![w2

wn, x] + [wl9 x]w2 "

wn.

Hence, this lemma can be proved by induction. LEMMA 28. Let Rbe a semi-prime ring and n a natural number v2" = 0 for all v E V, then V = 0.

If

Proof If v2 = 0 for all v E V, then for any JC E JR (ux + x*ι;) 2 = 0 so (vxf = 0. By Levitzki's lemma [5, Lemma 1.1], v=0 for all v E V. Assume that n > 1. For any v E V and x E 1?, we have ( ϋ ^ x - x ^ 2 " " 1 ) 2 " = 0 and hence (v2nlχ)2n+1 = 0. Applying Levitzki's lemma again and using the induction hypothesis, we conclude that V = 0. One might have noticed that the study of a symmetric subring U in R is based on the fact: If R is semi-prime, either U C Z + or U contains a nonzero ideal ofR. For a skew subgroup V, we have a parallel result for V2. LEMMA 29. If R is *-prime and [V 2 , V2] = 0, then V2CZ and [ V, V] = 0. Further, R satisfies the standard identity S[xu x29 *3, x 4 ] in 4 variables.

ON SUBRΪNGS OF RINGS WITH INVOLUTION

141

Proof. Consider first the situation when R is *~simple. If R = 2 A φ A * for some simple ring A, then K0=V=K, and so [ V\ V ] = 0 2 2 2 implies [A , A ] ~ 0 . Since A -A, R is also commutative, and the conclusions follow trivially. Assume that R is simple. Then V2CZ unless possibly 2JR = 0 and dimzjR = 4. Jf_JR is a_division ring, we have 1 1 1 2 2 x 2 xV^jr ==jcVjc*(χ- )*Vjr C V , so xV x~ QV for all x E R9 2 x^O. Hence V QZ by the Brauer-Cartan-Hua theorem. Suppose that R = F 2 for some field F with char.F = 2. If Z Π TV 0, say, α = α + α * E Z for some α£S,_then 1 = α-'α+(α" 1 α)*G Γ C V and hence JV C V. By Lemma 2, V is a symmetric subring. Since V = l V C V 2 , [ V, V] - 0 so V C Z by Theorem 4. If Z Π Γ - 0, then Z C S and * must be of transpose type, namely for some α E F . In this case. VC S = j Γ0

11 Λ

any

^ Γ0 £T,

.

L

1] Γ a A L ,,

, E V.

ducts, we get ca' ~ ac'. 0

y-T-ίί ,

bΛ

,

,

=

,

,

|α, 6,c E FK Since

, _ Γ0 commutes with

11 Γ a1 bΊ f L/ J for

Comparing the (1, l)-entries of the pro-

An argument like that in Theorem 4 shows

*1|6EF}.

Hence V2 = Z. Thus we have V2QZ

always. By Lemma 28, there exists v E V such that v2^0 provided O. Then v is invertible. Further, v~ι = v~ι(- v)(v~1)* E V, so C Z and V C Zϋ. Consequently [ V, V] = 0. Now assume that i? is *-ρrime and V^ 0. By Lemmas 1 and 28, Φ 09 so we may consider the quotient ring Q= |α E 1?, a EZ+, a^ 0}. O can be equipped with * by defining (a/a2)* = a*la2. Then Q is *-prime and V = {v/a2<= Q \v E V} is a skew subgroup of O If there is a nonzero *-ideal I of Q such that j n W 0 , then / C 5 ( O ) and hence [/,/] = 0. By the corollary to Lemma 6, Q is commutative and we are done. Suppose that / Π V ^ 0 for any nonzero *-ideal / of O Since / Π V contains an element a such that a4& Z and α 8 ^ 0 by Lemmas 1 and 28, and a8 is invertible, we have J = Q. In other words, Q is *-simple and so Vf2CZ(Q) and [ V\ Vf] - 0. Hence V2 C Z(i?) and [ V, V] = 0. Since Ko C V, we have [i^0, ^o] = 0 and hence R satisfies •S4[*i, JC2, Jt3, JC4] by Amitsur's Theorem [1]. __ We are now in a position to prove a series of theorems concerning V2. Since the proofs are parallel to those for U, we shall omit them unless some modification is needed. 30. If R is *-simple and V^ 0, then eitheτΎ2 = R orΨ2 is a field contained in Z + . THEOREM

142

PJEK-HWEE LEE

Proof. By Lemmas 1, 27 an

2

either V =JR or invertible elements aJΞ V2. Similarly, V2. Thus a'1 = 2

THEOREM

31.

If R is prime or *-prime, so is V .

THEOREM

32.

If R is semi-prime, then

THEOREM

33.

If R is semi-prime, so is V2.

THEOREM

34.

Sβ( V5) = V 2 Π $(J?).

THEOREM

35.

7/1? has no nil ideal other than 0, neither does V2.

THEOREM

36.

9?( V5) = V2 Π 9ί(i?).

THEOREM

37.

// JR ftαs no nonzero locally nilpotent ideals, neither

THEOREM

38.

S( V5) = V2 Π 2(i?).

THEOREM

39.

//

THEOREM

40.

/ / J R is semi-simple, so is V .

THEOREM

41.

%(v~2) = V2 n

2

z(V )CZ{R).

2

does V .

JR

is primitive or *-primitive, so is V2 provided

WO.

6GΓ. b)QV2.

2

It suffices to show that if a G V2 and a°b =0= b°a then The argument used in Theorem 30 shows that (1 + b)V\l + (The formal use_of the symbol 1 is all right.) Then b = + α 2 )(l + 6 ) G V2.

THEOREM 42. 7/1? is semi-primary, primary, or completely primary, 2 so is V provided V-έ J(R).

In the example given in [13], 2R = 0 and 1 G R, so K 2 = S. Hence K need not be noetherian even Jf JR is a commutative noetherian domain. However, K2, as well as S, inherits Goldie conditions when R is semi-prime. The proof of the next theorem is based on Lanski's argument [10] but is a little simpler. 2

THEOREM

43.

If R is a semi-prime Goldie ring, so is V2.

ON SUBRINGS OF RINGS WITH INVOLUTION

143

Proof. Since the a.c.c. on__right annihilators is inherited by sub2 rings, it suffices to show that V has PC infinite direct^ sum of nonzero 2 right ideals. Let {ρa} be a set of right ideals of_V jsuch that Σaρa 2 is direct. Denote by / the ideal of R generated by [ V , V2]. Then ΣaρJ is a direct sum__£f right ideals of R, so ρal = θ and hence paQ 2 2 V Π Ann./ C Z ( V ) for almost all a. Being a commutative semi-prime 2 subring of a Goldie ring, Z(V ) is itself a Goldie ring and hence pa = 0 for almost all α. Let R = F 2 , where F is a field with char.F = 2 and * is given by transpose. In this case, f = Ko = { ί* potent ideal j

α, Z> E Ff possesses the nil-

α E F even though R is simple. This example

kills the hope for f or Ko to inherit those nice properties we have discussed so far. Fortunately, the behavior of K is not that bad. THEOREM 44. If R is *-simple, either K = R or K is a commutative *-simple ring provided K^ 0.

Proof If char./? = 2, then K = S and hence the assertion follows from Theorem 4. Assume that chaτ.R^l. If [K2, K2] ^^_ then K a]so contains_the nonzero *-ideal of R generated by [K2,K2], so K = R. If K2 is commutative, then K2 C Z + by Theorem 30. Suppose that Z £ S, then a* ^ a for some α 6 Z , so β = a - α * ^ 0 . Thus, Sβ~xQK and hence S C JKΓ/8. Therefore, j R = S + ί C K Next, assume that Z C S. Then i? must be simple. By Lemma 29, 1? satisfies an identity of degree 4 and hence dim z l? S 4 by Kaplansky's Theorem. If R is a division ring, choose α 6 K , α ^ 0 , then KaΓ1 CK2Q Z. So KCZaC K, that is, K = Za. Hence K = Z(a) is a field. If R = F2 for some field F, the commutativity of K forces * to be of transpose type, say, ,

K = 11 while

if

,

α, fe E Ft.

- σ = π2

i

for

α6F

llπa

a J

and

J

=

,

. f o r

some σ E F.

Then

If - σ is not a square in F, K is a field; some

π E F,

X = LjφL2

L2 = i

U-πα

where

αEF

α

J|

are

Li = two

J

fields which are isomorphic via the map induced by *. 45. If R is *-prime, so is K. Proof If K} isjiot commutative, then K also contains the ideal generated by [K2, K2]. An argument exactly like t h a t j n Theorem 5 proves the *-primeness of K. Now we assume that K2 is a nonzero commutative ring. The quotient ring Q = {a I a \ a E i?, a E Z + , a/ 0} THEOREM

144

PJEK-HWEE LEE

is either a *-simple ring or a commutative *-prime ring relative to the involution (a/a)* - a* I a. In the former case, K(Q) is a commutative *-simple ring by the previous theorem. So in either case K(R) is contained in a commutative *-prime ring and hence is *-prime. LEMMA

46. // R is semi-prime, then CV(V2) = Z(V).

Proof. Assume first that R is *-prime. If _V^ is not commutative, then it contains a nonzero_jMdeal J of JR, so Cv(V2)CCR(I)CZ(R) by 2 2 2 Lemma 6 and hence Cy(V ) = Z(V). If [V , Vjj= 0, then V _C Z{R) and [V, V] = 0 by Lemma 29 and hence CΫ(V2)= V = Z(V). The semi-prime case can be built up easily via subdirect sum. The next lemma is crucial in the study of K. 47. Let R be a semi-prime ring and I a * -ideal of K. If = 0, then 1 = 0.

LEMMA

IΠK

Proof. If / Π K = 0, then JJZ S. For any α G J and k E K, ak = 2 (ak)* = - ka. Hence IC CR{K ) = Z(K) by Lemma 46. Thus IK C 2 / ΓΊ K = 0, so 7K = 0, and in particular I = 0. For any a E I and JC E 1?, we have α(jc-jc*) = θ, that is, ax = ax* and hence αxα = a(xa)* = 2 α x* = 0. Since JR is semi-prime, it follows that / = 0. LEMMA

48. // JR is semi-prime, and k EJC with kKk = 0, tfien

k =0. Proo/. For any JC G JR, fc(χ-χ*)fc=0 so kxk = kx*k. Then kxkxk = k(xkx*)k = 0 and hence /ci? is nil of index 3. So, fc = 0 by Levitzki's lemma. THEOREM

49. // R is semi-prime, so is K.

Proof Let / be a *-ideal of K such that I2 = 0. For any a G / Π K, 2 we have αKα C / = 0 so a = 0 by Lemma 48. Lemma 47 shows / = 0, so K has no nonzero nilpotent *-ideal and hence is semi-prime. THEOREM

50. // JR has no nil ideal other than 0, neither does K.

Proof. Let / be the ideal of R which^ is generated Jby [K ,K2]. Then 9i(/) = 0 and / C K If a G WJC) Π K and b G K2, then a2b_-ba2Ein$l(K) = 0. Thus a2EZ(K2) and by Lemma 46 a2EZ(K). But JSΓ is semi-prime and a is_nilpotent, so α 2 = 0 for all a E yi(K) Π K. In view of Lemma 28, $l(K) ΠK = 0 because $l(K) is itself a semi-prime ring. Hence, it follows from Lemma 47 that $l(K) = 0. 2

ON SUBRINGS OF RINGS WITH INVOLUTION

145

A similar argument proves the following THEOREM

51. If R has no nonzero locally nilpotent ideal, neither

does K. The proof of the next theorem is exactly like that of Theorem 39. THEOREM

52. If R is *-primitiυe, so is K provided K^O.

THEOREM

53. // R is semi-simple, so is K.

Proof. Let a G %(K) Π K. For any x 6 J ? , w e have ax o ( - ax*) = a(x - x* - xax*)G %(K)K C%(K). Hence aR is quasi-regular, so a = 0. By Lemma 47, $ ( £ ) = 0. THEOREM

54. If R is semi-prime artinian, so is K.

Proof. Immediate from Theorem 44. Unlike S, the semi-prime assumption on R is not sufficient to get the 2 converse theorems for K or K . For example, let F be a field with 2 char.F^ 2, σ an automorphism on F with σ = 1, and A a commutative semi-prime algebra over _F Put J R = F 0 Λ and define (a,a)* = {a\ a). Then K = F and K 2 = F σ are fields provided σ ^ 1, while R is not even *-prime. Further, if A possesses an identity and dimFA = oo? then R is neither artinian nor Goldie. On the other hand, the *-primeness is sufficient for our purpose. To begin with, we prove a lemma which is analogous to Lemma 3. 3t

55. Let R be a *-prime ring and I a nonzero '-ideal of R such that IΠKl = 0. If K0^0, then 1 = 0. LEMMA

Proof. If / Π Kl = 0, then (/ Π KQ)2 = 0. Since / is itself a semiprime ring, and / Π Ko is a skew subgroup of /, so / Π Ko = 0 by Lemma 28. Hence I CS. For any a E I and x E R, we have ax = (ax)* = x*α. So if α,fcG J and x G K, then αfcx = αx*b = xαb = abx*. That is, /2K0 = 0. Since R is *-prime and Ko ^ 0, it follows 7 = 0. 56. Let R be a *-prime ring and e the identity of K or If e ^ 0, then it is the identity of R.

LEMMA 2

V.

146

PJEK-HWEE LEE

Proof, Since the only nonzero central symmetric idempotent in a *-prime ring_te the identity, it suffices to show that e E Z(R). If e is the identity of V2, then ex - xe EL V2 for all x E R because V2 is a Lie ideal. If e works for K, then ex - xe = e(x - x*) + (ex* - xe) E K for all x E R. Hence e(ex - xe) = ex - xe — (ex - xe)e and this implies that eEZ(R). On the basis of Lemma 55, we can prove the converse theorems by using an argument parallel to that for U. 57. identity, so is R, THEOREM

THEOREM

58.

2

// R is *-prime, and K or V is a *-simple ring with 2

IfR is *-prime, and K or V is * -primitive, so is JR.

59. Let R be a *-prime ring and * not the identity 2 If K or V is semi-simple, so is R.

THEOREM

map.

2

1

2

2

2

Proof Since 3(V ) = V D^(R), so %(R) n K 0 = 0 if V is semisimple. By Lemma 55, R must be also semi-simple. In case K is 2 semi-simple, so is K by Theorem 41, and hence R is also semi-simple. 2

60. // R is ""-prime, and K or V has no nil ideal other than 0, then neither does R. THEOREM

2

61. If R is *-prime, and K or V has no nonzero locally nilpotent ideal, then neither does R. THEOREM

We close this paper with two theorems on chain conditions. THEOREM 62^ Let R be a *-prime ring. If * is not the identity map 2 and either K or V is artinian, then so is R.

Proof By Theorems 31 and 45, both K and V2 are *-prime. Say, if K is artinian, then it is^*-simple with identity, so R is also *-simple by Theorem 57 and hence K = R or K is commutative by Theorem 44. In the later case, R satisfies a polynomial identity, and is finite dimensional over a field contained in Z. Hence, R is artinian. The situation when V2 is artinian is the same. For a E R, let rR(a) = {x E R\ax = 0} be the right annihilator of a in R. Denote by #(/?) the right singular ideal of R, that is, {a E R I rR(a) Π p^ 0 for any nonzero right ideal p of R}. THEOREM

R.

63. Let R be a * -prime ring. If V2 is a Goldie ring, so is

ON SUBRINGS OF RINGS WITH INVOLUTION

147

Proof. If R is commutative, then Q = {a I a \ a E R, a E 5, a ^ 0} is a commutative *-simple ring, and hence R is a Goldie ring. Assume that JR is not commutative, while [ V\ V2] = 0. Then V2 C Z + and Q = {α /α I α E JR, α E Z + , t* ^ 0} is a *-simple ring. Since [ V, V] = 0, it follows that Q satisfies a polynomial identity, and hence is artinian. So, 2 2 JR is a Goldie ring. Lastly,_assume that [ V , V ) ^ 0 and let I be the 2 2 ideal of R generated by {V , V ]. Suppose {ρa} is a set^of right ideals of R which forms a direct sum. Then pj Cρa Π IC V2 and pj = 0 for almost all α. Consequently pa = 0 for almost all a. Consider #(JR)n/. If a Eβ(R)ni then for any nonzero right ideal p of /, plVO, so rR(a)Π pI^O and hence rj(α)Πpτ^0. In other words, $(jR)ίΊ IC$(I) = 0 because / is itself a semi-prime Goldie ring. So

REFERENCES 1. S. A. Amitsur, Identities in rings with involution, Israel J. Math., 7 (1969), 63-68. 2. W. E. Baxter and W. S. Martindale III, Rings with involution and polynomial identities, Canad. J. Math., 20 (1968),- 465-473. 3. J. Dieudonne, On the structure of unitary groups. Trans. Amer. Math. S o c , 72 (1952), 367-385. 4. I. N. Herstein, Lie and Jordan systems in simple rings with involution, Amer. J. Math., 78 (1956), 629-649. 5. I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1966. 6. I. N. Herstein, Noncotnmutative Rings, Cams Monograph, 15, Math. Assn. Amer., 1968. 7. N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloquium Publication 37, 1964. 8. N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloquium Publication 39, 1968. 9. C. Lanski, On the relationship of a ring and the subring generated by its symmetric elements, Pacific J. Math., 44 (1973), 581-592. 10. C. Lanski, Chain conditions in rings with involutions, J. London Math. Soc, to appear. 11. W. S. Martindale III, Rings with involution and polynomial identities, J. Algebra, 11 (1969), 186-194. 12. S. Montgomery, Lie structure of simple rings of characteristic 2, J. Algebra, 15 (1970), 387-407. 13. K. R. Nagarajan, Groups acting on noetherian rings, Niew Arch. Wiskunde, 16 (1968), 25-29. Received August 30, 1974 and in revised form December 6, 1974, This paper is based on a portion of the auther's Ph.D. Thesis at the University of Chicago under the supervision of Professor I. N. Herstein. UNIVERSITY OF CHICAGO Current address: NATIONAL TAIWAN UNIVERSITY TAIPEI, TAIWAN

PACIFIC JOURNAL OF MATHEMATICS EDITORS RICHARD ARENS

(Managing Editor)

J. DUGUNDJI

University of California Los Angeles, California 90024

Department of Mathematics University of Southern California Los Angeles, California 90007

R. A. BEAUMONT

D . GlLBARG AND J. MlLGRAM Stanford University Stanford, California 94305

University of Washington Seattle, Washington 98105

ASSOCIATE EDITORS E F. BECKENBACH

B. H. NEUMANN

R WOLF

K. YOSHIDA

SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY

UNIVERSITY OF SOUTHERN CALIFORNIA STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON

AMERICAN MATHEMATICAL SOCIETY

The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. Mathematical papers intended for publication in the Pacific Journal of Mathematics should be in typed form or offset-reproduced (not dittoed), double spaced with large margins. Underline Greek letters in red, German in green, and script in blue. The first paragraph or two must be capable of being used separately as a synopsis of the entire paper. Items of the bibliography should not be cited there unless absolutely necessary, in which case they must be identified by author and Journal, rather than by item number. Manuscripts, in duplicate, may be sent to any one of the four editors. Please classify according to the scheme of Math. Reviews, Index to Vol. 39. All other communications should be addressed to the managing editor, or Elaine Barth, University of California, Los Angeles, California, 90024. 100 reprints are provided free for each article, only if page charges have been substantially paid. Additional copies may be obtained at cost in multiples of 50. The Pacific Journal of Mathematics is issued monthly as of January 1966. Regular subscription rate: $72.00 a year (6 Vols., 12 issues). Special rate: $36.00 a year to individual members of supporting institutions. Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 103 Highland Boulevard, Berkeley, California, 94708. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION Printed at Jerusalem Academic Press, POB 2390, Jerusalem, Israel. Copyright © 1975 Pacific Journal of Mathematics All Rights Reserved

Pacific Journal of Mathematics Vol. 60, No. 2

October, 1975

Waleed A. Al-Salam and A. Verma, A fractional Leibniz q-formula . . . . . . . . . . . . . Robert A. Bekes, Algebraically irreducible representations of L 1 (G). . . . . . . . . . . . Thomas Theodore Bowman, Construction functors for topological semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen LaVern Campbell, Operator-valued inner functions analytic on the closed disc. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonard Eliezer Dor and Edward Wilfred Odell, Jr., Monotone bases in L p . . . . . . Yukiyoshi Ebihara, Mitsuhiro Nakao and Tokumori Nanbu, On the existence of global classical solution of initial-boundary value problem for cmu − u 3 = f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Gordon, Unconditional Schauder decompositions of normed ideals of operators between some l p -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gary Grefsrud, Oscillatory properties of solutions of certain nth order functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irvin Roy Hentzel, Generalized right alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . Zensiro Goseki and Thomas Benny Rushing, Embeddings of shape classes of compacta in the trivial range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emil Grosswald, Brownian motion and sets of multiplicity . . . . . . . . . . . . . . . . . . . . . Donald LaTorre, A construction of the idempotent-separating congruences on a bisimple orthodox semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pjek-Hwee Lee, On subrings of rings with involution . . . . . . . . . . . . . . . . . . . . . . . . . . Marvin David Marcus and H. Minc, On two theorems of Frobenius . . . . . . . . . . . . . Michael Douglas Miller, On the lattice of normal subgroups of a direct product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grattan Patrick Murphy, A metric basis characterization of Euclidean space . . . . . Roy Martin Rakestraw, A representation theorem for real convex functions . . . . . . Louis Jackson Ratliff, Jr., On Rees localities and Hi -local rings . . . . . . . . . . . . . . . . Simeon Reich, Fixed point iterations of nonexpansive mappings . . . . . . . . . . . . . . . . Domenico Rosa, B-complete and Br -complete topological algebras . . . . . . . . . . . . Walter Roth, Uniform approximation by elements of a cone of real-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmut R. Salzmann, Homogene kompakte projektive Ebenen . . . . . . . . . . . . . . . . . . Jerrold Norman Siegel, On a space between B H and B∞ . . . . . . . . . . . . . . . . . . . . . . Robert C. Sine, On local uniform mean convergence for Markov operators . . . . . . James D. Stafney, Set approximation by lemniscates and the spectrum of an operator on an interpolation space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Árpád Száz, Convolution multipliers and distributions . . . . . . . . . . . . . . . . . . . . . . . . . Kalathoor Varadarajan, Span and stably trivial bundles . . . . . . . . . . . . . . . . . . . . . . . . Robert Breckenridge Warfield, Jr., Countably generated modules over commutative Artinian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Yuan, On the groups of units in semigroups of probability measures . . . . . . . .

1 11 27 37 51

63 71 83 95 103 111 115 131 149 153 159 165 169 195 199 209 217 235 247 253 267 277 289 303

ON SUBRINGS OF RINGS WITH INVOLUTION P JEK -H WEE L EE

Vol. 60, No. 2

October 1975

PACIFIC JOURNAL OF MATHEMATICS Vol. 60, No. 2, 1975

ON SUBRINGS OF RINGS WITH INVOLUTION PJEK-HWEE L E E

We give a systematic account on the relationship between a ring R with involution and its subrings S and K, which are generated by all its symmetric elements or skew elements respectively.

I. Introduction. Let R be a ring with involution * and 5 the subring generated by the set S of all symmetric elements in R. The relationship between R and S has been studied by various authors. In [3] Dieudonne showed that if R is a division ring of characteristic not 2, then either S = R or SQZ(R), the center of R. Later Herstein [4] extended this result by proving S = R for any simple ring R with dim z i?>4 and char.i?^2. The restriction on characteristic was removed by Montgomery [12]. Recently, Lanski [9] proved that if_R is prime or semi-prime, so is 5. In §2 of this paper, we show that S can inherit a number of ring-theoretic properties such as primitivity, semisimplicity, absence of nonzero nil ideals etc.. In doing so, a notion called symmetric subring, which is a generalization of S and its *homomorphic images, is introduced so that a group of theorems of the same type, including Lanski's results, can be proved via a more or less unified argument. We show also that numerous radicals of S are merely the contractions from those of R. As a consequence, we see that R modulo its prime radical behaves much like S in many respects. In §3 we establish a corresponding theory for K, the subring generated by all skew elements. The only result hitherto known concerning K was as follows [4], [12]: If R is_simple and dimzi? >4, then K - RL As a matter of fact, the subring K2 is more closely related to JR than K is. We apply thejtechnique developed in §2 to study the relationship between R and K2, and then derive some parallel theorems for K. II. Symmetric subrings. Our work depends heavily on the notion of Lie ideals. By a Lie ideal U of R we mean an additive subgroup which is invariant under all inner derivations of JR. That is, [uy x] = ux - xu E U for all u E U and x E R. The following lemma concerning Lie ideals will be referred to frequently in the sequel, and it is a combination of some results in [5]. LEMMA 1. Let Rbe a semi-prime ring and U a subring and Lie ideal of R. Then U contains the ideal of R which is generated by [U,U]. If U is commutative, then u2G Z for all u E U. 131

132

PJEK-HWEE LEE

Rings with involution abound with examples of Lie ideals. One can easily show that any subring, generated by symmetric elements and containing T = {x + x*|JC G R} the set of all traces, must be a Lie ideal. In particular, both S and T are Lie ideals. Another essential property of S follows from the next lemma. We denote by N the set of all norms, i.e. N = {xx*| JC G R}. LEMMA

andxUx*C

2. Let U be an additive subgroup of R such that T C U C S Ufor all x G R. IfN C 17, thenxUx*Q U for all x G R.

Proof. We prove by induction that xux unx * G Ό for all x G R and Mi, , un G (7. The case n = 1 is clear. Assume the assertion holds for n - 1 then XM!W2

UnX * = | > , Mj] [U2

Mn, X * ] + ( X M ^ *>M2

Un + UX(XU2

MnX * )

- u1xx*u2- - - un E U because C7 is a Lie ideal. A subring U of R is called a symmetric subring if: U is generated by a set of symmetric elements. TUNCU xUx*QU for all x <Ξ R.

DEFINITION.

1. 2. 3.

In light of Lemma 2, we know that S is a symmetric subring. From now on, U will always denote a symmetric subring of R. We call an ideal I of R a *-ideal if /* = / . LEMMA 3. // R is semi-prime and I is a *-ideal of R such that I Π U = 0, ί/ien 7 = 0.

Proof. For any a G /, α 2 =

Then / is nil of

Recall that a ring JR is called a *-simple ring if JR2 φ 0 and R has no *-ideal other than 0 and JR. It is well-known that JR is *-simple if and only if either R is simple or JR = A 0 A * for some simple ring A [8, p. 14]. Let Z + = Z Π S. Then if JR is *-simple, we have Z + = 0 or Z + is a field. THEOREM 4. contained in Z + .

// JR is *-simple, then either U = R or U is a field

Proof If U is not commutative, by Lemma 1 it contains a nonzero *-ideal of R so U = JR. Assume that [£/, t/] = 0; then U C S. In this

ON SUBRINGS OF RINGS WITH INVOLUTION

133

case, we need only to prove [/CZ, for if u E U and u^ 0 then If R = A φ A * for some simple ring A, then T = U = S. Thus [C/, I/] = 0 implies [A, A ] = 0 and so R is commutative. If 1? is simple, then U, being a commutative subring and Lie ideal of R, must be central unless 2R = 0 and dimzjR = 4 [5, Theorem 1.5]. So let us examine all possible 4-dimensional cases. If R is a division ring, then x~ιUx = x~ι{xUx*)x = Ux*x CU for all x E i? with x ^ 0. Hence U C Z by the Brauer-Cartan-Hua theorem [7, Theorem 7.13.1,-Cor.]. There remains the case R = F2 where F is a field with char.F = 2. We claim that * must be of symplectic type. Assume the contrary,

[c

[ab

d\

d

for some a E F with ά = α, where - denotes the * induced automorphism on F. Thus

For any α E F, we have Γ0

Lo

l

o J L o oj U o

so a = a. Next, if | , [ab Γ ft [a + c

+d 01Γ0

a + aΛ\a

cj

£ t/ then

0 1 Γ α b U O 0 1 ΓO O l Γ α 6 1 b\ [ab c J L l Oj [ l OJ [ab c_Γ

and hence α = c. But if I a,

1 £ I/, then

Γα 01 Γl O l Γ α bUl 01 LO OJ L0 O j [ a b a \ [ θ OJ yields α = 0 . S o l / = T = | • u• cause T is not a subring. T

,

U

^ f t G F ί which is ridiculous be-

^ \ \a Consequently,

ucs-{[°

U

bT \d , =

bλ

, and b

] c

a M

134

For

PJEK-HWEE LEE

\a

any

bλ

L

\a'

b'Λ^TJ

,

;

.

ε l / , we have

\a

bλ\a'

,

b Ί ^

.

, E r [/τ and

1

hence be = b'c by comparing the diagonal entries of the product. If there exists \a',

b

U with b'/O, then

,]e

ι

where a - c'b'~ .

However, O 01

ΓO 01 Γα'

' Oj Ll o J L c '

bΊTO

;

a

forces ί? = 0, a contradiction. Hence U C (| ILc other hand, if ΓO

cl

ΓO l Ί Γ α

[O OJ L θ O J L c

01

α ' J L l Q\

\\α,c E F\. 1 αj J

0 1 Γ 0 11

αJLo O J

On the

e U

implies c = 0. Therefore, U CZ. Following [11], we say R is *-prime if the product of any two nonzero *-ideals is still not zero. It is easy to see that J? is *-ρrime if and only if aRb = a *Rb = 0 implies a = 0 or b = 0. As a consequence, any + nonzero element in Z is regular in a *-prime ring R. We remind the reader of of a well-known fact that a nonzero Lie ideal of a semi-prime ring always contains elements with nonzero square. THEOREM

5.

If R is *-prime, so is U.

Proof. If [U,U]τ^ 0, then U contains a nonzero *-ideal / of R. For any two *-ideals A, B of U with AB = 0, we have IAIB C AB = 0, so either IAI = 0 or B = 0, ending up with A = 0 or B = 0. Assume that u y o while [U,ί/] = 0. By Lemma 1, there exists u0SU such that UQ E Z but Mo ^ 0. So consider the ring Q of fractions a/a with α E R and α E Z Π t/, α ^ 0. Q is also *-prime with respect to the involution given by (a/a)* = a*/α, and [/' = {u/α E Q) u E [/} is a symmetric subring of Q. As a matter of fact, Q is *-simple. For if J is any nonzero *-ideal of O, / Γl [/V 0 and hence (v/βf^O for some v/β EJΠU'. Since v2EZ, v/β is invertible and so / - Q. By the

ON SUBRINGS OF RINGS WITH INVOLUTION

135

+

previous theorem, U' CZ (Q) and hence U is an integral domain contained in Z+(R). Let CR(V) = {x G R \xυ = vx for all vEV}be the centralizer of a set V in R. LEMMA 6. Let j y 0 be an ideal (or *-ideal) of a prime (resp. *-prime) ring JR. Then CR(I) C Z

/. For a E I, b E CR (I) and JC E i?, we have abx = feαx = αxfe, or equivalently, α(bx - xb) = 0. That is, /[C R (/),R] = 0. Hence [C R (/),K] = 0 and so CR(I)CZ. COROLLARY. Let R be a prime (or *-prime) ring and I a nonzero ideal (resp. *-ideal) ofR such that [I, I] = 0. Then R is commutative. THEOREM

7.

If R is semi-prime, then

Z(U)QZ(R).

Proof Assume first that R is *-prime. If [U, U] = 0, then Z(U) = UCZ(R) by Theorem 5. If [£/, f/]^0, then U contains a nonzero *-ideal / of R, so Z( U) C C* (I) C Z(l?) in view of Lemma 6. In either case, [Z(U),R] = 0. Now assume that R is semi-prime; then R is a subdirect sum of *-prime rings πa(R). Since πa(U) is a symmetric subring of πa(R% we know [ττa(Z(U)\ πa(R)] C [Z(τrα(17)), π β (Λ)] = 0 for all a. Hence, [Z( [/), R ] = 0. The same reduction to *-prime rings together with Theorem 5 gives an alternate proof for Lanski's theorem: THEOREM

8.

If R is semi-prime, so is U.

With this established, we are able to consider the relationship between the prime radicals ?β(R) and ^(U). THEOREM 9.

$ ( 17) = 17 n W

)•

Proof Since U/[UΠ^(R)] - [17 + S$(R)]/ψ(R) which is a symmetric subring of the semi-prime ring JR/^3(jR), so U/[U Π?β(R)] is semi-prime by Theorem 8 and hence ^(U)CUΠ %$(R). On the other hand, if a E U Γ) ^5(1?), then a E U and any m -system in JR containing a must contain 0. [7, Theorem 8.2.3]. Certainly, any m -system in U containing a contains 0. That is, a E^(U). It is well-known that a ring without nonzero nil ideals is a subdirect sum of rings with the following property [6, p. 53]: There exists a nonnilpotent element a such that an(I) E I for all nonzero ideal I

136

PJEK-HWEE LEE

One can impose this condition only on the *-ideals and show that it is a hereditary property. Then, making use of subdirect sum decomposition, we can prove that U inherits the freedom from nonzero nil ideals. Instead of doing this way, we prefer to present a direct proof by considering the nil radical $l(U) of U. THEOREM

10. // R has no nil ideal other than 0, neither does U.

Proof. Let I be the ideal of JR which is generated by [ 17,17]. Since JR possesses no nonzero nil ideal, neither does J, considered as a ring. Hence 9l(U) Π / = 0. For any a G %l(U) and u G [/, we have [a, u] E 9l(U) Π / - 0. Thus 9l(U) C Z(U). Since U is semi-prime by Theorem 8, As an immediate consequence, we have THEOREM 11.

31 (U) = U Π 9l(R ).

Proceed as above with "locally nilpotent" in place of "nil" and with Levitzki radical S in place of 9t, we get THEOREM 12. If R has no nonzero locally nilpotent ideal, neither does 17. THEOREM 13.

£([/) = U Π £ ( # ) .

In [2] the notion of *-primitive ring was introduced as a ring admitting a *-faithful irreducible module M (i.e. Mr = Mr* = 0 implies r = 0). One can easily verify that a ring is *-primitive if and only if it is either primitive or a subdirect sum of a primitive ring and its opposite with the exchange involution. We know that a nonzero ideal of a primitive ring is itself primitive. The proof is applicable to the following more general fact. LEMMA 14. Let R be a primitive (or *-primitive) ring. Suppose that I is a nonzero ideal (resp. *-ideal) of JR, and A is a subring (resp. *-subring, i.e. A* = A) containing I. Then A is also primitive (resp. ^-primitive). THEOREM 15.

If R is primitive or *-primitive, so is U.

Proof If [ 17, U] 7^ 0, U contains a nonzero *-ideal of JR, so it is primitive or *-ρrimitive by Lemma 14. Assume that U is commutative. Then 17 C Z + and every element in JR is quadratic over

ON SUBRINGS OF RINGS WITH INVOLUTION

137

+

Z . Hence R satisfies a polynomial identity. According to Kaplansky's theorem [6, Theorem 6.3.1], R is *-simple and hence U is a field by Theorem 4. Using the fact that a semi-simple ring is a subdirect sum of *-primitive rings, we get immediately THEOREM

16. If R is semi-simple, so is U.

In fact, the semi-simplicity of 5 was first proved by Herstein. His elegant proof was the inspiration of our next theorem which relates the Jacobson radicals of R and U. THEOREM

17. %(U)=UΠ

%(R).

Proof. For a E $(L/) and x E JR, we have ax oax* = ax + ax* + axax* = a(x + x* +

xax*)e%(U)UC%(U).

Thus aR is quasi-regular and hence a GU n $(!?). Conversely, if α ε i / Π S C R ) , α ° & = 0 for some 6 e Λ, then & = 6°(α°fc)* = (f)ofe*)oα*G I/. That is, UΠ^(R) is a quasi-regular ideal of U, so With Theorem 17 in hand, we are ready to study some non-semisimple rings. Following [7], we say R is semi-primary, primary, or completely primary according as R/%(R) is an artinian, simple artinian, or division ring respectively. Since U/$(U) is isomorphic to a symmetric subring of R/J(R)? by Theorem 4 we have THEOREM

18. // R is primary or completely primary, so is U.

As to semi-primary rings, we need some information about the descending chain condition. In a paper [10] which is to appear, Lanski proved that if JR is artinian and \ E R, then so is S. For our purpose, we prove LEMMA

19. If R is semi-prime artinian, so is U.

Proof. By the Wedderburn-Artin theorem, we may write R = -RiΘ'" * ®Rn where each Rt is *-simple. Denote by e, the identity of jRi, then ex E Z + and so eJJex is a symmetric subring of Rx for each /. By Theorem 4, each eJJe, is artinian, so is U = eJJex 0 φ enUen. THEOREM

20. // JR is semi-primary, so is U.

138

PJEK-HWEE LEE

We remark that the assertion corresponding to Lemma 19 for ascending chain condition is not true even if R is a commutative integral domain. A counter example can be found in [13]. Let 9ϊ stand for any of the four radicals Sβ, 2, 9? and $. We have shown 9t(I7) = 17 n 91(1?). If 9t(ϊ7) = U, then C/C9»(JR), so 0 is a symmetric subring of the semi-prime ring R/ΐR(R), and hence ?R(R) = R by Lemma 3. That is, if U is locally nilpotent, nil or quasi-regular, so is R. On the other hand, 91(17) = 0 need not imply 91(1?) = 0. For example, let 1? = F + A be the algebra obtained by adjunction of an identity to a trivial algebra A over a field F with char. Fφ 2. Define (a + a)* - a - a for a E F and a E A. Then S - F is a field, while 9t(JR) = A is a nilpotent ideal. In case A has infinite dimension, this example shows also that JR is not artinian although S is. However, we still have some results on 9Ϊ(JR). For if 91(17) = 0, then the *-ideal 9ΐ(l?) has trivial intersection with 17, hence is nil of index 2. Thus we have aRa = 0 for any a E 9ΐ(l?) and consequently 91(1?) = Besides, U is isomorphic to a symmetric subring of Realizing this fact, one might not be surprised to see that ), instead of 1? itself, satisfies the same properties as U does. 21. Let R be a semi-prime ring and e the identity of Then e is also the identity of JR.

LEMMA

U.

Proof By Theorem 7, eEZ(U)CZ(R). Since e E S, 1 = {x - ex \x E JR} is a *-ideal of 1?. If a - ea E 17, then a — eα = β(α - eα) = 0. Thus / Π [/ = 0 and so / = 0. In other words, e is the identity of JR. The case when JR is semi-prime and S is simple was thoroughly studied by Lanski [9]. An example was given there that 1? is an integral domain but not simple while S is. In the presence of an identity, we have THEOREM 22. Let R be a semi-prime ring. If U is a *-simple ring with identity, so is R.

Proof Let I be any nonzero *-ideal of 1?. Then / Π U^ 0, and the ^simplicity of U implies UCl By Lemna 21, U contains the identity of JR, SO / = 1?. Even if U is a field, 1? can be semi-prime but not simple. The simplest example is the direct sum of two copies of a field with the exchange involution. This example illustrates why we deal with only *-primeness and *-primitivity in what follows.

ON SUBRINGS OF RINGS WITH INVOLUTION

139

THEOREM 23. (1) If U is semi-prime, ψ(R) is nil of index 2. (2) // U is *-prime, so is R/^(R)

Proof We have proved (1) in the discussion before Lemma 21. As to (2), we may assume without loss of generality that R is semiprime. Let / and / be *-ideals of R such that I/ = 0. Then (/ Π U)(J Π U) = 0, so / n U = 0 or / Π U = 0, ending up with / = 0 or J = 0. Suppose that JR is a *-prime ring and I a nonzero *-ideal of R. If / possesses a *-faithful irreducible module M, write M = ml for some m E M and m ^ 0, and define a map from M x R into M by sending (ma, r) to m(ar). One can easily check that such a map is well defined and that M becomes a *-faithful irreducible R -module. This is the content of 24. Let R be a *-prime ring and I a nonzero ideal of R. If I is ""-primitive, so is R. LEMMA

THEOREM 25. (1) // U is semi-simple, then %(R) = ^(R) is nil of index 2. (2) // U is *-primitive, so is R/^(R)

Proof We have seen the proof of (1) earlier. As to (2), we assume that R is semi-prime. By Theorem 23, JR is *-prime. If [U, U]^0, then U contains a nonzero *-ideal / of JR. Lemma 14 shows that I is itself *-primitive and hence JR is also *-primitive by the previous lemma.* If U is commutative, it is *-simple with identity. It follows from Theorem 22 that R is ""-primitive. THEOREM

26. // U is semi-primary, so is R.

Proof It suffices to show that if R is semi-prime and U is artinian, then JR is also artinian. In this case, we have U - UXQ) 0 Un, where each Ui is *-simple artinian. Let e, be the identity of L/i; then eieZ(U)CZ(R). Since 1 = ex+ + en, R=R1φ'"®Rn, with Ri = eR. Each Rt is then semi-prime and contains C7, as a symmetric subring. By Theorem 22 Rt is *-simple, so either Lζ = Ri or L/j is a field. If Ui is a field, then R< satisfies a polynomial identity and hence is a finite dimensional algebra over a field contained in Z(JRr). In either case, Ri is always artinian. Hence R must be also artinian. III. Subrings generated by skew elements. In contrast to S, K is not necessarily a Lie ideal of i?. For instance, in F2 with

140

PJEK-HWEE LEE

char. F^2 and transpose as *, ^ = if "b

A \a,beFγ Although

[Λ ] ro L-i

- 1 1 r o l i p on r i o i r o η o J L - i oJLo o j Lo o J t - i o j

falls outside of K. However, both {x - x * I x E JR}, are always Lie ideals.

K2

and

where

Kl,

Ko =

DEFINITION. By. a skew subgroup V of R we mean a subgroup of R such that K0Q VQK and xVx* C V for all x E JR. Henceforth we shall use V to stand for a skew subgroup of R without further explanation. LEMMA 27.

Proof

V2 is a Lie ideal of JR.

For vu v2 E V and x E R, we have 2

[VιV2, X] = ϋi(lλ>* + X * ϋ 2 ) - ( ^ i X * + Xϋi)ϋ 2 E V .

If Wi,

, wn E V2 and x E R, then [>!

wn,x] = W![w2

wn, x] + [wl9 x]w2 "

wn.

Hence, this lemma can be proved by induction. LEMMA 28. Let Rbe a semi-prime ring and n a natural number v2" = 0 for all v E V, then V = 0.

If

Proof If v2 = 0 for all v E V, then for any JC E JR (ux + x*ι;) 2 = 0 so (vxf = 0. By Levitzki's lemma [5, Lemma 1.1], v=0 for all v E V. Assume that n > 1. For any v E V and x E 1?, we have ( ϋ ^ x - x ^ 2 " " 1 ) 2 " = 0 and hence (v2nlχ)2n+1 = 0. Applying Levitzki's lemma again and using the induction hypothesis, we conclude that V = 0. One might have noticed that the study of a symmetric subring U in R is based on the fact: If R is semi-prime, either U C Z + or U contains a nonzero ideal ofR. For a skew subgroup V, we have a parallel result for V2. LEMMA 29. If R is *-prime and [V 2 , V2] = 0, then V2CZ and [ V, V] = 0. Further, R satisfies the standard identity S[xu x29 *3, x 4 ] in 4 variables.

ON SUBRΪNGS OF RINGS WITH INVOLUTION

141

Proof. Consider first the situation when R is *~simple. If R = 2 A φ A * for some simple ring A, then K0=V=K, and so [ V\ V ] = 0 2 2 2 implies [A , A ] ~ 0 . Since A -A, R is also commutative, and the conclusions follow trivially. Assume that R is simple. Then V2CZ unless possibly 2JR = 0 and dimzjR = 4. Jf_JR is a_division ring, we have 1 1 1 2 2 x 2 xV^jr ==jcVjc*(χ- )*Vjr C V , so xV x~ QV for all x E R9 2 x^O. Hence V QZ by the Brauer-Cartan-Hua theorem. Suppose that R = F 2 for some field F with char.F = 2. If Z Π TV 0, say, α = α + α * E Z for some α£S,_then 1 = α-'α+(α" 1 α)*G Γ C V and hence JV C V. By Lemma 2, V is a symmetric subring. Since V = l V C V 2 , [ V, V] - 0 so V C Z by Theorem 4. If Z Π Γ - 0, then Z C S and * must be of transpose type, namely for some α E F . In this case. VC S = j Γ0

11 Λ

any

^ Γ0 £T,

.

L

1] Γ a A L ,,

, E V.

ducts, we get ca' ~ ac'. 0

y-T-ίί ,

bΛ

,

,

=

,

,

|α, 6,c E FK Since

, _ Γ0 commutes with

11 Γ a1 bΊ f L/ J for

Comparing the (1, l)-entries of the pro-

An argument like that in Theorem 4 shows

*1|6EF}.

Hence V2 = Z. Thus we have V2QZ

always. By Lemma 28, there exists v E V such that v2^0 provided O. Then v is invertible. Further, v~ι = v~ι(- v)(v~1)* E V, so C Z and V C Zϋ. Consequently [ V, V] = 0. Now assume that i? is *-ρrime and V^ 0. By Lemmas 1 and 28, Φ 09 so we may consider the quotient ring Q= |α E 1?, a EZ+, a^ 0}. O can be equipped with * by defining (a/a2)* = a*la2. Then Q is *-prime and V = {v/a2<= Q \v E V} is a skew subgroup of O If there is a nonzero *-ideal I of Q such that j n W 0 , then / C 5 ( O ) and hence [/,/] = 0. By the corollary to Lemma 6, Q is commutative and we are done. Suppose that / Π V ^ 0 for any nonzero *-ideal / of O Since / Π V contains an element a such that a4& Z and α 8 ^ 0 by Lemmas 1 and 28, and a8 is invertible, we have J = Q. In other words, Q is *-simple and so Vf2CZ(Q) and [ V\ Vf] - 0. Hence V2 C Z(i?) and [ V, V] = 0. Since Ko C V, we have [i^0, ^o] = 0 and hence R satisfies •S4[*i, JC2, Jt3, JC4] by Amitsur's Theorem [1]. __ We are now in a position to prove a series of theorems concerning V2. Since the proofs are parallel to those for U, we shall omit them unless some modification is needed. 30. If R is *-simple and V^ 0, then eitheτΎ2 = R orΨ2 is a field contained in Z + . THEOREM

142

PJEK-HWEE LEE

Proof. By Lemmas 1, 27 an

2

either V =JR or invertible elements aJΞ V2. Similarly, V2. Thus a'1 = 2

THEOREM

31.

If R is prime or *-prime, so is V .

THEOREM

32.

If R is semi-prime, then

THEOREM

33.

If R is semi-prime, so is V2.

THEOREM

34.

Sβ( V5) = V 2 Π $(J?).

THEOREM

35.

7/1? has no nil ideal other than 0, neither does V2.

THEOREM

36.

9?( V5) = V2 Π 9ί(i?).

THEOREM

37.

// JR ftαs no nonzero locally nilpotent ideals, neither

THEOREM

38.

S( V5) = V2 Π 2(i?).

THEOREM

39.

//

THEOREM

40.

/ / J R is semi-simple, so is V .

THEOREM

41.

%(v~2) = V2 n

2

z(V )CZ{R).

2

does V .

JR

is primitive or *-primitive, so is V2 provided

WO.

6GΓ. b)QV2.

2

It suffices to show that if a G V2 and a°b =0= b°a then The argument used in Theorem 30 shows that (1 + b)V\l + (The formal use_of the symbol 1 is all right.) Then b = + α 2 )(l + 6 ) G V2.

THEOREM 42. 7/1? is semi-primary, primary, or completely primary, 2 so is V provided V-έ J(R).

In the example given in [13], 2R = 0 and 1 G R, so K 2 = S. Hence K need not be noetherian even Jf JR is a commutative noetherian domain. However, K2, as well as S, inherits Goldie conditions when R is semi-prime. The proof of the next theorem is based on Lanski's argument [10] but is a little simpler. 2

THEOREM

43.

If R is a semi-prime Goldie ring, so is V2.

ON SUBRINGS OF RINGS WITH INVOLUTION

143

Proof. Since the a.c.c. on__right annihilators is inherited by sub2 rings, it suffices to show that V has PC infinite direct^ sum of nonzero 2 right ideals. Let {ρa} be a set of right ideals of_V jsuch that Σaρa 2 is direct. Denote by / the ideal of R generated by [ V , V2]. Then ΣaρJ is a direct sum__£f right ideals of R, so ρal = θ and hence paQ 2 2 V Π Ann./ C Z ( V ) for almost all a. Being a commutative semi-prime 2 subring of a Goldie ring, Z(V ) is itself a Goldie ring and hence pa = 0 for almost all α. Let R = F 2 , where F is a field with char.F = 2 and * is given by transpose. In this case, f = Ko = { ί* potent ideal j

α, Z> E Ff possesses the nil-

α E F even though R is simple. This example

kills the hope for f or Ko to inherit those nice properties we have discussed so far. Fortunately, the behavior of K is not that bad. THEOREM 44. If R is *-simple, either K = R or K is a commutative *-simple ring provided K^ 0.

Proof If char./? = 2, then K = S and hence the assertion follows from Theorem 4. Assume that chaτ.R^l. If [K2, K2] ^^_ then K a]so contains_the nonzero *-ideal of R generated by [K2,K2], so K = R. If K2 is commutative, then K2 C Z + by Theorem 30. Suppose that Z £ S, then a* ^ a for some α 6 Z , so β = a - α * ^ 0 . Thus, Sβ~xQK and hence S C JKΓ/8. Therefore, j R = S + ί C K Next, assume that Z C S. Then i? must be simple. By Lemma 29, 1? satisfies an identity of degree 4 and hence dim z l? S 4 by Kaplansky's Theorem. If R is a division ring, choose α 6 K , α ^ 0 , then KaΓ1 CK2Q Z. So KCZaC K, that is, K = Za. Hence K = Z(a) is a field. If R = F2 for some field F, the commutativity of K forces * to be of transpose type, say, ,

K = 11 while

if

,

α, fe E Ft.

- σ = π2

i

for

α6F

llπa

a J

and

J

=

,

. f o r

some σ E F.

Then

If - σ is not a square in F, K is a field; some

π E F,

X = LjφL2

L2 = i

U-πα

where

αEF

α

J|

are

Li = two

J

fields which are isomorphic via the map induced by *. 45. If R is *-prime, so is K. Proof If K} isjiot commutative, then K also contains the ideal generated by [K2, K2]. An argument exactly like t h a t j n Theorem 5 proves the *-primeness of K. Now we assume that K2 is a nonzero commutative ring. The quotient ring Q = {a I a \ a E i?, a E Z + , a/ 0} THEOREM

144

PJEK-HWEE LEE

is either a *-simple ring or a commutative *-prime ring relative to the involution (a/a)* - a* I a. In the former case, K(Q) is a commutative *-simple ring by the previous theorem. So in either case K(R) is contained in a commutative *-prime ring and hence is *-prime. LEMMA

46. // R is semi-prime, then CV(V2) = Z(V).

Proof. Assume first that R is *-prime. If _V^ is not commutative, then it contains a nonzero_jMdeal J of JR, so Cv(V2)CCR(I)CZ(R) by 2 2 2 Lemma 6 and hence Cy(V ) = Z(V). If [V , Vjj= 0, then V _C Z{R) and [V, V] = 0 by Lemma 29 and hence CΫ(V2)= V = Z(V). The semi-prime case can be built up easily via subdirect sum. The next lemma is crucial in the study of K. 47. Let R be a semi-prime ring and I a * -ideal of K. If = 0, then 1 = 0.

LEMMA

IΠK

Proof. If / Π K = 0, then JJZ S. For any α G J and k E K, ak = 2 (ak)* = - ka. Hence IC CR{K ) = Z(K) by Lemma 46. Thus IK C 2 / ΓΊ K = 0, so 7K = 0, and in particular I = 0. For any a E I and JC E 1?, we have α(jc-jc*) = θ, that is, ax = ax* and hence αxα = a(xa)* = 2 α x* = 0. Since JR is semi-prime, it follows that / = 0. LEMMA

48. // JR is semi-prime, and k EJC with kKk = 0, tfien

k =0. Proo/. For any JC G JR, fc(χ-χ*)fc=0 so kxk = kx*k. Then kxkxk = k(xkx*)k = 0 and hence /ci? is nil of index 3. So, fc = 0 by Levitzki's lemma. THEOREM

49. // R is semi-prime, so is K.

Proof Let / be a *-ideal of K such that I2 = 0. For any a G / Π K, 2 we have αKα C / = 0 so a = 0 by Lemma 48. Lemma 47 shows / = 0, so K has no nonzero nilpotent *-ideal and hence is semi-prime. THEOREM

50. // JR has no nil ideal other than 0, neither does K.

Proof. Let / be the ideal of R which^ is generated Jby [K ,K2]. Then 9i(/) = 0 and / C K If a G WJC) Π K and b G K2, then a2b_-ba2Ein$l(K) = 0. Thus a2EZ(K2) and by Lemma 46 a2EZ(K). But JSΓ is semi-prime and a is_nilpotent, so α 2 = 0 for all a E yi(K) Π K. In view of Lemma 28, $l(K) ΠK = 0 because $l(K) is itself a semi-prime ring. Hence, it follows from Lemma 47 that $l(K) = 0. 2

ON SUBRINGS OF RINGS WITH INVOLUTION

145

A similar argument proves the following THEOREM

51. If R has no nonzero locally nilpotent ideal, neither

does K. The proof of the next theorem is exactly like that of Theorem 39. THEOREM

52. If R is *-primitiυe, so is K provided K^O.

THEOREM

53. // R is semi-simple, so is K.

Proof. Let a G %(K) Π K. For any x 6 J ? , w e have ax o ( - ax*) = a(x - x* - xax*)G %(K)K C%(K). Hence aR is quasi-regular, so a = 0. By Lemma 47, $ ( £ ) = 0. THEOREM

54. If R is semi-prime artinian, so is K.

Proof. Immediate from Theorem 44. Unlike S, the semi-prime assumption on R is not sufficient to get the 2 converse theorems for K or K . For example, let F be a field with 2 char.F^ 2, σ an automorphism on F with σ = 1, and A a commutative semi-prime algebra over _F Put J R = F 0 Λ and define (a,a)* = {a\ a). Then K = F and K 2 = F σ are fields provided σ ^ 1, while R is not even *-prime. Further, if A possesses an identity and dimFA = oo? then R is neither artinian nor Goldie. On the other hand, the *-primeness is sufficient for our purpose. To begin with, we prove a lemma which is analogous to Lemma 3. 3t

55. Let R be a *-prime ring and I a nonzero '-ideal of R such that IΠKl = 0. If K0^0, then 1 = 0. LEMMA

Proof. If / Π Kl = 0, then (/ Π KQ)2 = 0. Since / is itself a semiprime ring, and / Π Ko is a skew subgroup of /, so / Π Ko = 0 by Lemma 28. Hence I CS. For any a E I and x E R, we have ax = (ax)* = x*α. So if α,fcG J and x G K, then αfcx = αx*b = xαb = abx*. That is, /2K0 = 0. Since R is *-prime and Ko ^ 0, it follows 7 = 0. 56. Let R be a *-prime ring and e the identity of K or If e ^ 0, then it is the identity of R.

LEMMA 2

V.

146

PJEK-HWEE LEE

Proof, Since the only nonzero central symmetric idempotent in a *-prime ring_te the identity, it suffices to show that e E Z(R). If e is the identity of V2, then ex - xe EL V2 for all x E R because V2 is a Lie ideal. If e works for K, then ex - xe = e(x - x*) + (ex* - xe) E K for all x E R. Hence e(ex - xe) = ex - xe — (ex - xe)e and this implies that eEZ(R). On the basis of Lemma 55, we can prove the converse theorems by using an argument parallel to that for U. 57. identity, so is R, THEOREM

THEOREM

58.

2

// R is *-prime, and K or V is a *-simple ring with 2

IfR is *-prime, and K or V is * -primitive, so is JR.

59. Let R be a *-prime ring and * not the identity 2 If K or V is semi-simple, so is R.

THEOREM

map.

2

1

2

2

2

Proof Since 3(V ) = V D^(R), so %(R) n K 0 = 0 if V is semisimple. By Lemma 55, R must be also semi-simple. In case K is 2 semi-simple, so is K by Theorem 41, and hence R is also semi-simple. 2

60. // R is ""-prime, and K or V has no nil ideal other than 0, then neither does R. THEOREM

2

61. If R is *-prime, and K or V has no nonzero locally nilpotent ideal, then neither does R. THEOREM

We close this paper with two theorems on chain conditions. THEOREM 62^ Let R be a *-prime ring. If * is not the identity map 2 and either K or V is artinian, then so is R.

Proof By Theorems 31 and 45, both K and V2 are *-prime. Say, if K is artinian, then it is^*-simple with identity, so R is also *-simple by Theorem 57 and hence K = R or K is commutative by Theorem 44. In the later case, R satisfies a polynomial identity, and is finite dimensional over a field contained in Z. Hence, R is artinian. The situation when V2 is artinian is the same. For a E R, let rR(a) = {x E R\ax = 0} be the right annihilator of a in R. Denote by #(/?) the right singular ideal of R, that is, {a E R I rR(a) Π p^ 0 for any nonzero right ideal p of R}. THEOREM

R.

63. Let R be a * -prime ring. If V2 is a Goldie ring, so is

ON SUBRINGS OF RINGS WITH INVOLUTION

147

Proof. If R is commutative, then Q = {a I a \ a E R, a E 5, a ^ 0} is a commutative *-simple ring, and hence R is a Goldie ring. Assume that JR is not commutative, while [ V\ V2] = 0. Then V2 C Z + and Q = {α /α I α E JR, α E Z + , t* ^ 0} is a *-simple ring. Since [ V, V] = 0, it follows that Q satisfies a polynomial identity, and hence is artinian. So, 2 2 JR is a Goldie ring. Lastly,_assume that [ V , V ) ^ 0 and let I be the 2 2 ideal of R generated by {V , V ]. Suppose {ρa} is a set^of right ideals of R which forms a direct sum. Then pj Cρa Π IC V2 and pj = 0 for almost all α. Consequently pa = 0 for almost all a. Consider #(JR)n/. If a Eβ(R)ni then for any nonzero right ideal p of /, plVO, so rR(a)Π pI^O and hence rj(α)Πpτ^0. In other words, $(jR)ίΊ IC$(I) = 0 because / is itself a semi-prime Goldie ring. So

REFERENCES 1. S. A. Amitsur, Identities in rings with involution, Israel J. Math., 7 (1969), 63-68. 2. W. E. Baxter and W. S. Martindale III, Rings with involution and polynomial identities, Canad. J. Math., 20 (1968),- 465-473. 3. J. Dieudonne, On the structure of unitary groups. Trans. Amer. Math. S o c , 72 (1952), 367-385. 4. I. N. Herstein, Lie and Jordan systems in simple rings with involution, Amer. J. Math., 78 (1956), 629-649. 5. I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1966. 6. I. N. Herstein, Noncotnmutative Rings, Cams Monograph, 15, Math. Assn. Amer., 1968. 7. N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloquium Publication 37, 1964. 8. N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloquium Publication 39, 1968. 9. C. Lanski, On the relationship of a ring and the subring generated by its symmetric elements, Pacific J. Math., 44 (1973), 581-592. 10. C. Lanski, Chain conditions in rings with involutions, J. London Math. Soc, to appear. 11. W. S. Martindale III, Rings with involution and polynomial identities, J. Algebra, 11 (1969), 186-194. 12. S. Montgomery, Lie structure of simple rings of characteristic 2, J. Algebra, 15 (1970), 387-407. 13. K. R. Nagarajan, Groups acting on noetherian rings, Niew Arch. Wiskunde, 16 (1968), 25-29. Received August 30, 1974 and in revised form December 6, 1974, This paper is based on a portion of the auther's Ph.D. Thesis at the University of Chicago under the supervision of Professor I. N. Herstein. UNIVERSITY OF CHICAGO Current address: NATIONAL TAIWAN UNIVERSITY TAIPEI, TAIWAN

PACIFIC JOURNAL OF MATHEMATICS EDITORS RICHARD ARENS

(Managing Editor)

J. DUGUNDJI

University of California Los Angeles, California 90024

Department of Mathematics University of Southern California Los Angeles, California 90007

R. A. BEAUMONT

D . GlLBARG AND J. MlLGRAM Stanford University Stanford, California 94305

University of Washington Seattle, Washington 98105

ASSOCIATE EDITORS E F. BECKENBACH

B. H. NEUMANN

R WOLF

K. YOSHIDA

SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY

UNIVERSITY OF SOUTHERN CALIFORNIA STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON

AMERICAN MATHEMATICAL SOCIETY

The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. Mathematical papers intended for publication in the Pacific Journal of Mathematics should be in typed form or offset-reproduced (not dittoed), double spaced with large margins. Underline Greek letters in red, German in green, and script in blue. The first paragraph or two must be capable of being used separately as a synopsis of the entire paper. Items of the bibliography should not be cited there unless absolutely necessary, in which case they must be identified by author and Journal, rather than by item number. Manuscripts, in duplicate, may be sent to any one of the four editors. Please classify according to the scheme of Math. Reviews, Index to Vol. 39. All other communications should be addressed to the managing editor, or Elaine Barth, University of California, Los Angeles, California, 90024. 100 reprints are provided free for each article, only if page charges have been substantially paid. Additional copies may be obtained at cost in multiples of 50. The Pacific Journal of Mathematics is issued monthly as of January 1966. Regular subscription rate: $72.00 a year (6 Vols., 12 issues). Special rate: $36.00 a year to individual members of supporting institutions. Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 103 Highland Boulevard, Berkeley, California, 94708. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION Printed at Jerusalem Academic Press, POB 2390, Jerusalem, Israel. Copyright © 1975 Pacific Journal of Mathematics All Rights Reserved

Pacific Journal of Mathematics Vol. 60, No. 2

October, 1975

Waleed A. Al-Salam and A. Verma, A fractional Leibniz q-formula . . . . . . . . . . . . . Robert A. Bekes, Algebraically irreducible representations of L 1 (G). . . . . . . . . . . . Thomas Theodore Bowman, Construction functors for topological semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen LaVern Campbell, Operator-valued inner functions analytic on the closed disc. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonard Eliezer Dor and Edward Wilfred Odell, Jr., Monotone bases in L p . . . . . . Yukiyoshi Ebihara, Mitsuhiro Nakao and Tokumori Nanbu, On the existence of global classical solution of initial-boundary value problem for cmu − u 3 = f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Gordon, Unconditional Schauder decompositions of normed ideals of operators between some l p -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gary Grefsrud, Oscillatory properties of solutions of certain nth order functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irvin Roy Hentzel, Generalized right alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . Zensiro Goseki and Thomas Benny Rushing, Embeddings of shape classes of compacta in the trivial range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emil Grosswald, Brownian motion and sets of multiplicity . . . . . . . . . . . . . . . . . . . . . Donald LaTorre, A construction of the idempotent-separating congruences on a bisimple orthodox semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pjek-Hwee Lee, On subrings of rings with involution . . . . . . . . . . . . . . . . . . . . . . . . . . Marvin David Marcus and H. Minc, On two theorems of Frobenius . . . . . . . . . . . . . Michael Douglas Miller, On the lattice of normal subgroups of a direct product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grattan Patrick Murphy, A metric basis characterization of Euclidean space . . . . . Roy Martin Rakestraw, A representation theorem for real convex functions . . . . . . Louis Jackson Ratliff, Jr., On Rees localities and Hi -local rings . . . . . . . . . . . . . . . . Simeon Reich, Fixed point iterations of nonexpansive mappings . . . . . . . . . . . . . . . . Domenico Rosa, B-complete and Br -complete topological algebras . . . . . . . . . . . . Walter Roth, Uniform approximation by elements of a cone of real-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmut R. Salzmann, Homogene kompakte projektive Ebenen . . . . . . . . . . . . . . . . . . Jerrold Norman Siegel, On a space between B H and B∞ . . . . . . . . . . . . . . . . . . . . . . Robert C. Sine, On local uniform mean convergence for Markov operators . . . . . . James D. Stafney, Set approximation by lemniscates and the spectrum of an operator on an interpolation space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Árpád Száz, Convolution multipliers and distributions . . . . . . . . . . . . . . . . . . . . . . . . . Kalathoor Varadarajan, Span and stably trivial bundles . . . . . . . . . . . . . . . . . . . . . . . . Robert Breckenridge Warfield, Jr., Countably generated modules over commutative Artinian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Yuan, On the groups of units in semigroups of probability measures . . . . . . . .

1 11 27 37 51

63 71 83 95 103 111 115 131 149 153 159 165 169 195 199 209 217 235 247 253 267 277 289 303

*When life gives you a hundred reasons to cry, show life that you have a thousand reasons to smile*

© Copyright 2015 - 2021 PDFFOX.COM - All rights reserved.