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Self-adjoint ideals in baer *-ringsGary F. Birkenmeier a & Jae Keol Park ba Department of Mathematics, University of Southwestern Louisiana,Lafayette, LA, 70504-1010, U.S.A E-mail:b Department of Mathematics, Busan National University, Busan,609-735, South Korea E-mail:Published online: 27 Jun 2007.

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COMMUNICATIONS IN ALGEBRA, 28(9), 42594268 (2000)

SELF-ADJOINT IDEALS I N B A E R *-RINGS

Gary F. Birkenmeier Department of Mathematics

University of Southwestern Louisiana Lafayette, LA 70504-1010

U. S. A. E-mail: gf bll27Qusl. edu

and

Jae Keol Park Department of Mathematics Busan National University

Busan 609-735, South Korea E-mail: j kparkQhyowon . cc . pusan. ac . kr

0. Introduct ion. Throughout this paper R denotes an associative ring with unity. If an involution * is defined on R, then R is called a *-ring. If X is a nonempty subset of R, then X is said to be self-adjoint if X = X*. We use M,(R) for the n x n matrix ring over R. The complex number field, the rational number field, and the ring of integers are denoted by @, Q, and Z, respectively.

Recall from [14], R is a Baer ring if the right annihilator of every nonempty subset is generated, as a right ideal, by an idempotent. If R is a *-ring, then R is called a Baer *-ring if the right annihilator of every nonernpty subset is generated, as a right ideal, by a projection (i.e., e is a projection if e = e2 and e* = e). The study of Baer and Baer *-rings has its roots in functional analysis [14] and [I]. The class of Baer *-rings includes the von Neumann algebras (e.g., the algebra of all bounded operators on a Hilbert space), the commutative Cc-algebra C(T) of continuous complex valued functions on a Stonian space T, and the complete *-regular rings. Also note that the Sherman-Takeda theorem in [18] and [19] shows that every CL-algebra has a universal enveloping von Neumann algebra (hence a Baer *-ring). Kaplansky shows in [14] that the definitions of a Baer ring and a Baer *-ring are left-right symmetric.

Copyright O 2000 by Marcel Dekker, Inc.

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From [a], a ring R is quasi-Baer if the right annihilator of every right ideal is generated, as a right ideal, by an idempotent. This is a nontrivial general- ization of the class of Baer rings. For example prime rings with nonzero right singular ideal [15] are quasi-Baer and not Baer. If R is a non-Priifer com- mutative domain, then Mn(R) (n > 1) is a prime P I quasi-Baer ring which is not a Baer ring [16] and [14, p.171. The n x n (n > 1) upper triangular matrix ring over a domain which is not a division ring is a quasi-Baer ring but not Baer [16] and [14, p.161. Every semiprime right FPF ring [lo, P. 1681 and every piecewise domain [ll] and [5, Corollary 4.131 are quasi-Baer rings.

We define R to be a quasi-Baer *-ring if the right annihilator of every ideal is generated, as a right ideal, by a projection. In [a], Clark proves that the definition of a quasi-Baer ring is left-right symmetric. This is shown in the sequel for quasi-Baer *-rings. There are numerous examples of quasi- Baer *-rings which are not Baer *-rings. For example, from [6] Mz(C)[x] is a quasi-Baer *-ring (* is the conjugate transpose involution) which is not a Baer *-ring. Other examples will be provided in the sequel.

It is well known that every ideal in a von Neumann algebra is self-adjoint. This follows from the polar decomposition of elements [9, p.91. In [I , p.1341, an abstract polar decomposition is defined (also see [14]) for elements of a *- ring. If a *-ring has this polar decomposition property (PD) for all elements, then every ideal is self-adjoint [l, Exercise 5A, p.1351. In [I, pp.135-1361 several conditions on Baer *-rings are given which imply PD. Handelman [12, Theorem 3.61 shows that if R is a commutative *-ring such that Mn(R) is a Baer *-ring for all n, then every ideal of R is self-adjoint. Also a routine argument shows that in a *-regular ring (i.e., a regular ring with a proper involution) every ideal is self-adjoint. At this point, one might conjecture that at least in a commutative Baer *-ring, every ideal is self-adjoint. However we will show by example that this is not the case. This motivates one to ask: How closely is the ideal structure of a (quasi-) Baer *-ring connected to its *-ideal (i.e., self-adjoint ideal) structure?

In Section 1, our main result (Theorem 1.8) shows that every ideal in a quasi-Baer *-ring is "essentially sandwiched" between self-adjoint ideals. This result is applied in Section 2 to show that minimal left ideals of a quasi- Baer *-ring have either a "Hermitian" form or an "alternating" form. Since not every ideal in a (quasi-) Baer *-ring is self-adjoint, it is natural to ask if the (quasi-) Baer *-condition can be characterized in terms of self-adjoint ideals. An affirmative answer is provided in Section 3.

Recall from [2] an idempotent e E R is left (resp. right) semicentral in R if eRe = Re (resp. eRe = eR). Equivalently, an idempotent e is left (resp, right) semicentral in R if eR (resp. Re) is an ideal of R. Note if R is semiprime then every left (resp. right) semicentral idempotent is central. For a nonempty subset X of R , 1 ( X ) and r ( X ) denote the left and right annihilators of X, respectively. Also Ze(R) and Z,(R) denote the left and

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SELF-ADJOINT IDEALS IN BAER *-RINGS 426 1

right singular ideals of R, respectively. The right socle of R will be symbolized by Soc(R). Observe that if R is semiprime, then Soc(R) coincides with the left socle of R. If X is a nonempty subset of R , then X Ce R (resp. X C, R) denote that X is a left (resp. right) ideal of R , respectively. For two left (resp. right) ideals X and Y of R, X C;SS Y (resp. X CyS Y) denote that X is left (resp, right) essential in Y, respectively. Recall 11, p.101 an involution * is called proper if x'x = 0 implies x = 0. Other terminology can be found in [I].

1. Essentially self-adjoint. In this section, we present our main result which shows that every ideal in a quasi-Baer *-ring is an essential extension of a self-adjoint ideal and is also essential in a self-adjoint ideal (i.e., "essentially sandwiched").

Proposition 1.1. Let R be a *-ring. Then the following conditions are equivalent:

(i) R is a quasi-Baer *-ring. (ii) R is a quasi-Baer ring in which each left semicentral idempotent is a

projection. (iii) R is a semiprime quasi-Baer ring in which each central idempotent is

a projection (iv) The left annihilator of every left ideal is generated, as a left ideal, by

a projection.

Proof. (i)=+(ii) Clearly R is a quasi-Baer ring. Let e be a left semicentral idempotent. Then r( (1 - e)R) = eR. There exists a projection f such that eR = f R. Hence eR is a self-adjoint ideal. So Re* = eR. Thus e := ee* = e*. Consequently e is a projection.

(ii)$(iii) Since every central idempotent is a left semicentral idempotent, we only need to show that R is semiprime. Assume x E R such that xRx = 0. Then x E r(xR) = eR, where e is a left semicentral idempotent. Then eR is a self-adjoint ideal of R. Hence ex* = x*. So xe = x. But xe == 0. Thus R is semiprime.

(iii)+(iv) Let Y be a left ideal of R. Since R is quasi-Baer, there exists a right semicentral idempotent such that l (Y) = Re. Since Re is an ideal of R , er = ere for every r E R and (1 - e)Re is an ideal of R. The semiprimeness of R implies that (1 - e)Re = 0. Thus re = ere for every r E R. So we have er = ere = re for every r E R. Therefore e is central and hence e is a projection.

(iv)=+(i) Let X be a right ideal of R. Observe that ( r (X))* = e(X*). There exists a projection e such that C(X*) = Re. Hence r ( X ) = eR. Therefore R is a quasi-Baer *-ring.

Corollary 1.2. If R is a prime ring with an involution *, then R is a quasi-Baer *-ring.

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Proof. Since a prime ring is a quasi-Baer ring, this result follows from Proposition 1.1 (iii).

Example 1.3. There exists a semiprime Baer ring R with an involution * which is not a quasi-Baer *-ring. Take R = F $ F, where F is a field and * is defined by (a, b)* = (b, a) , for all a, b E F.

The next two examples show that, in general, not every ideal of a Baer *-ring or a quasi-Baer *-ring is self-adjoint.

Example 1.4. There exists a commutative Baer *-ring R which has an ideal I such that I is not self-adjoint. Let * be the involution on C[x] induced by the conjugate involution on C. Take cu E C which is not a real. Then the difference between x - a and its conjugate is a nonzero element of C. Hence the ideal I = (x - a)C[x] is not self-adjoint. Note that @[XI is a Banach @-algebra.

Surprisingly, by [12, Theorem 3.61, the n x n matrix ring, M,(@[[x]]), is a Baer *-ring (the involution is conjugate transpose) in which every ideal is self-adjoint for all n. The next example is a prime subring of M2(@[[x]]) which further illustrates the delicate nature of the condition: every ideal is self-adjoint.

Example 1.5. Let R = ( @[[xll xc [ [x l l ) mom 14, Example A] and x@[[xIl @[[XI]

Corollary 1.2, R is a quasi-Baer *-ring which is not a Baer ring. But the

ideal I = ( x@[[xll xc[[xll) is not self-adjoint, x2@@[[41 x@I[xIl

However all is not lost as our main theorem will demonstrate. Let X, Y be ideals of R such that X c Y. We say X is ideal essential in Y if any nonzero ideal of R which has nonzero intersection with Y also has nonzero intersection with X .

Lemma 1.6. Let R be a semiprime ring with ideals X and Y. If X is ideal essential in Y, then r (X) = r(Y) = 1(X) = C(Y).

Proof. Clearly, r(Y) 5 r (X) . Observe, if Ynr (X) # 0, then X n r ( X ) # 0, a contradiction. Hence Y fl r (X) = 0. So r (X) E r(Y). Thus r (X) = r (Y). Since R is semiprime, r (X) = C(X) = C(Y).

Lemma 1.7. Let R be a semiprime ring with an involution * and I a nonzero ideal of R. Then the following conditions are equivalent:

(i) r ( I ) is a self-adjoint ideal of R. (ii) 11* # 0. (iii) 11* Ce,sS I. (iv) 11* zys I. Proof. (i)+(ii) Assume 11* = 0. Then I* C r ( I ) . Hence I c r ( I ) , a

contradiction.

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SELF-ADJOINT IDEALS IN BAER *-RINGS 4263

(ii)+(iii) First we claim that I n I* gYs I . Let Y 2, R such that I n I* n Y = 0 and Y C I . Since Y ( I n I * ) = 0 and R is semiprime, it follows that RY E r ( I n I*) . Also the right annihilator of a self-adjomt ideal in a semiprime ring is self-adjoint, thus (RY)* E r ( I n I*). So R Y + (RY)* is a self-adjoint ideal of R contained in I. Hence RY + (RY)* I n I* n r ( I n I * ) = 0. Thus Y = 0 and I n I* C Y I . Now we will show that 11* c Y s I n I * . Let K ST R such that 11* n K = 0. Then I . l*RK = 0. Observe ( I*RK)2 C I I * R K = 0. Hence I * R K = 0. But R K c I n I * , hence (RK)2 C I ( R K ) = 0. So R K = 0, hence K = 0. Therefore 11* CFSS I .

(iii)+(iv) Since 11* C_YS I, then 11* # 0. Now use a proof atlalogous to (ii) + (iii).

(iv)+(i) By Lemma 1.6, r ( I I * ) = r ( I ) . Since R is semiprime and 11* is a self-adjoint ideal of R, it follows that r ( I I * ) is a self-adjoint ideal of R.

Theorem 1.8. Let R be a quasi-Baer *-ring. (i) If I is an ideal of R, then r ( I + I * ) = r ( I ) = r ( I I* ) = eR, where e is

a central projection. (ii) If I is a nonzero ideal of R , then there is a central projection f E R

such that I T * Ce,SS I CYS I + I* STS f R , and

(iii) If X is a nonzero right ideal, then X X * # 0. (iv) If Y is a nonzero left ideal, then Y*Y # 0.

Proof. (i) If I = 0, we are finished. So assume I # 0. By Proposition 1.1 there exists a central projection e such that r ( I + I * ) = eR. Then r ( I ) = eR@Y, where Y = (1 -e )Rnr ( I ) . Observe that Y* [(I*). By Proposition 1.1, R is semiprime. So [(I*) = r ( I*) . Hence Y n Y* C eR n Y = 0. Since r(Y) is self-adjoint, Lemma 1.7 yields Y = 0. Hence r ( I + I * ) = r ( I ) . Since r ( I ) is a self-adjoint ideal, we obtain r ( I I* ) = r ( I ) from Lemrnas 1.6 and 1.7.

(ii) From Lemma 1.7, 11* EyS I . By [3, Lemma 2.21, I CpS jr R, where f is a central idempotent. From Proposition 1.1, f is a projection. Since f R is a self-adjoint ideal, I + I* CYS f R. A similar argument using the fact that [(I) = r ( I ) yields 11* CTs I Cys I + I* Cess -, f R .

(iii) Assume X X * = 0. hen X * & r ( X ) = r (RX). By part; (i) , r ( R X ) is a self-adjoint ideal. Hence X r (RX) , a contradiction.

(iv) This part is similar to part (iii).

Corollary 1.9. Let R be a quasi-Baer *-ring. The following ideals are self-adjoint: (i) annihilator ideals; (ii) minimal ideals; (iii) Soc(R).

Proof. Part (i) is a consequence of Theorem 1.8(i). From Theorem l.8(ii), each minimal ideal I = 11* = I * . From part (iii), observe that in a semiprime ring every homogeneous component is a minimal ideal and tht: right socle equals the left socle of R.

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In [7, Example 4.41 Brown provides an example of a group algebra R = K G over a field K , which is a uniform primitive ring with Z,(R) # 0. Now R is a ring with involution defined by ( x a g g ) ' = xagg- ' . Therefore by Corollary 1.2, R is a quasi-Baer *-ring. Since (Z,(R))* = Zt(R), Theorem 1.8 yields that Zr(R)Ze(R) ce,8s R and Z,(R)Ze(R) Gtss R. Moreover in the introduction of [7], Brown conjectures that if R = K G is a group algebra over a field K , then Z,(R) = Ze(R). Since R has the involution indicated above, our next result supports this conjecture by showing that for group algebras which are quasi-Baer *-rings the left and right second singular ideals coincide.

Corollary 1.10. Let R be a quasi-Baer *-ring. Then the left and the right second singular ideals coincide. In particular R is left nonsingular if and only if R is right nonsingular.

Proof. Since (Z,(R))* = Ze(R), the result is a consequence of Theorem 1.8(ii).

We close this section with the following open problem: Characterize the Baer *-rings (quasi-Baer *-rings) in which every ideal is self-adjoint.

2. Minimal left ideals. In this section we show that a semiprime quasi- Baer ring R has a ring decomposition R = A $ B , where Soc(A) is essential in A and Soc(B) = 0. Moreover each homogeneous component of A is a primitive ring and a direct summand of R. Next by considering quasi- Baer *-rings, we are able to show that the minimal left ideals have either a "Hermitian" form or an "alternating" form. Although the results of this section are left-right symmetric, we consider left ideals to conform to the usage in [17] which is the main reference for this section.

Lemma 2.1. Let R be a quasi-Baer ring with a prime ideal P. Then exactly one of the following conditions is satisfied:

(i) P is left essential in R. (ii) P = Re for some idempotent e E R.

Proof. Assume that there exists 0 # x E R such that P n Rx = 0. Then Rx C r ( P ) = dR, where d = d2 E R. Then P C R ( l - d). Also (1 - d)Rd = 0 E P. Since d @ P, then 1 - d E P. Let e = 1 - d. Then P = Re.

Observe that from Corollary 1.9(i), if R is a quasi-Baer *-ring and P is a nonself-adjoint prime ideal, then P is left and right essential in R.

Lemma 2.2. Let R be a quasi-Baer ring with a minimal left ideal of the form Re, where e = e2. Then R = A @ B (left ideal direct sum), where A is a primitive ring with unity, Soc(A) # 0, and B is a primitive ideal of R.

Proof. From the hypothesis R = Re $ R ( l - e), where R ( l - e) is a maximal left ideal. If R ( l - e) contains no nonzero ideal, then R is a left primitive ring with nonzero socle, so take R = A. Otherwise there exists a

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SELF-ADJOINT IDEALS IN BAER *-RINGS 4265

nonzero left primitive ideal B of R such that B 2 R ( l - e). From Lemma 2.1, B = Rb for some idempotent b E R. Let a = 1 - b. Then A = Ra is a left primitive ring with unity a. Since A is ring isomorphic to R / B , then A has an isomorphic copy of Re. Hence Soc(A) # 0. Note that A has nonzero socle, then A is both left and right primitive.

Proposition 2.3. Let R be a semiprime quasi-Baer ring. Then R = A $ B (ring direct sum), where Soc(A) is right and left essential in A and Soc(B) = 0. Moreover, if H is a homogeneous component of A, there exists a central idempotent c E A such that H is in cA and cA is a primitive ring.

Proof. From (3, Lemma 2.21, there exists a central idempotent (1. E R such that Soc(R) is right and left essential in aR. Let A = a R and B - (1 - a)R. Then Soc(A) = Soc(R) and Soc(B) = 0. The remainder of the p:roof follows from the semiprimeness of A and Lemma 2.2.

Note that in Proposition 2.3 if there are only finitely many nonisomorphic minimal left ideals in R , then A is a finite direct sum of primitive rings.

In the remaining results of this section, Proposition 2.3 allows us to con- sider each minimal left ideal of a quasi-Baer *-ring R as being contained in a primitive *-ring which is a direct summand of R. Primitive *:-rings with minimal one sided ideals are discussed in detail in [13]. For the terminology in the remainder of this section see [17, pp.302-3031.

Proposition 2.4. Let R be a quasi-Baer *-ring. If L is a minim.al left ideal of R , then either:

(i) L is generated by a projection; or (ii) EndR(,!,) is a field and xx* = 0 for all x E L.

Proof. Except for minor adjustments, the proof is similar to 117, Propo- sition 2.13.19, p.3011. Observe that condition L"L = 0 in [17, Proposition 2.13.19, p.3011 is eliminated by Theorem 1.8.

Observe that in a Baer *-ring only Proposition 2.4(i) holds because every idempotent generated left ideal is generated by a projection.

Corollary 2.5. Let R be a quasi-Baer *-ring. If L is a minimal left ideal of R , then either (viewing L as a right vector space over D = Endjs(L)):

(i) L has a (*)-compatible Hermitian form which is not alternating; or (ii) L has a (*)-compatible alternating form and D is a field.

Proof. Except for minor adjustments, the proof is similar to [17, Theorem 2.13.21, p.3021.

The next example shows that for a quasi-Baer *-ring both forms may appear.

Example 2.6. [17, Example 2.13.22, p.3031 Let R = M,(F), where F is a field and n > 1. Then R is a prime ring. By Corollary 1.2, R is a quasi- Baer *-ring for any involution *. From [17, Example 2.13.22, p.3031 the

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transpose involution will yield that L = Rell has a (*)-compatible Hermitian form which is not alternating. Note that for this involution if R is a Baer *-ring for all n , then F must be formally real [12, Corollary 2.51. If the symplectic involution is used for n an even positive integer, then L = Rell has a (*)-compatible alternating form and D is a field. Since this involution is improper, then R is not a Baer *-ring. Hence R is a Baer ring which is a quasi-Baer *-ring but not a Baer *-ring.

3. Characterizations in t e r m s of t h e *-structure. In this section we characterize Baer *-rings and quasi-Baer *-rings in terms of their *- structures.

Proposi t ion 3.1. Let R be a *-ring. Then the following conditions are equivalent:

(i) R is a Baer *-ring. (ii) * is a proper involution and the right annihilator of every nonempty

self-adjoint subset is generated, as a right ideal, by a projection. (iii) * is a proper involution and the left annihilator of every nonempty

self-adjoint subset is generated, as a left ideal, by a projection.

Proof. (i)+(ii) By [14, Theorem 2.11, * is a proper involution. The re- mainder of the proof of this implication follows from the definition of a Baer *-ring.

(ii)+(i) Let Y be a nonempty subset of R. Form Y*Y = { y ' y 1 y E Y). Then Y'Y is a nonempty self-adjoint subset of R. Then there exists a projection e such that r(Y*Y) = eR. Let y E Y, then we have y * y e = 0. By [I, Proposition 1, p.101, ye = 0. Hence r(Y*Y) r(Y). So r(Y) = r(Y*Y) = eR. Therefore R is a Baer *-ring.

(i)*(iii) The proof of this equivalence is analogous to the proof of (i)*(ii).

Example 3.2. There exists a Baer ring R which is a quasi-Baer *-ring with a proper involution, but R is not a Baer *-ring. Let R = M2(Z) with the involution * being transposition. By [14, p.171, R is a Baer ring. By Corollary 1.2, R is a quasi-Baer *-ring. From 112, Proposition 1.51, R is not a Baer *-ring. But since Mz(Q) is a Baer *-ring, * is a proper involution on R.

One might conjecture that Y U Y' could be used for the self-adjoint nonempty subset in the proof of (ii)+(i) in Proposition 3.1. The next exam- ple eliminates this possibility.

Example 3.3. Let R = Mz(Z3), where Z3 is the field of three elements. By

[14, p.391, R is a Baer *-ring. Let Y = { ( i)}. T h e n r ( Y ~ Y * ) = O #

r (Y) = r(Y*Y).

Proposi t ion 3.4. Let R be a *-ring. The following conditions are equivalent: (i) R is a quasi-Baer *-ring.

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SELF-ADJOINT IDEALS IN BAER +-RINGS 4267

(ii) aRa* # 0 for all 0 # a E R and the right annihilator of every self- adjoint ideal is generated, as a right ideal, by a projection.

(iii) aRa* # 0 for all 0 # a E R and the left annihilator of every self-adjoint ideal is generated, as a left ideal, by a projection.

Proof. (i)+(ii) Theorem 1.8(iii) yields the condition aRa* # 0 for all 0 # a E R. The remainder of the proof of this implication follows from the definition of a quasi-Baer *-ring.

(ii)=+(i) First assume K is a self-adjoint ideal of R such that K2 = 0. Then K r(K) = eR for some projection e E R. Hence 0 = Ke and so 0 = e*K* = eK. Therefore K = eK = 0. By 117, Proposition 2.13.34, p.3071, R is semiprime. Now let I be a nonzero ideal of R. Then II* # 0. From Lemma 1.6 and Lemma 1.7(iii), r ( I ) = r(II*) . Since II* is self-adjoint, r ( I ) is generated by a projection.

( i ) ~ ( i i i ) The proof of this equivalence is analogous to the proof of (i)@(ii).

Example 1.3 shows that the condition aRa* # 0 for all 0 # a: E R is not superfluous in Proposition 3.4. Note that Example 2.6 and Mz(Z2) provide examples of quasi-Baer *-rings with an improper involution, where Z2 is the field of two elements.

ACKNOWLEDGEMENTS The authors wish to thank the referee for his/her helpful comments and

suggestions. The second author was supported in part by Korea Research Foundation with Research Grant Project No.1998-001-D00006. Also the sec- ond author appreciates the kind hospitality of the University of Southwestern Louisiana during his stay.

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19. Z. Takeda, Conjugate spaces of operator algebras, Proc. Japan Acad. 30 (1954), 90-95.

Received: April 1999

Revised: August 1999

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