JOURNAL OF ALGEBRA 187, 548᎐578Ž. 1997 ARTICLE NO. JA966796

Mininjective Rings*

W. K. Nicholson†

Department of Mathematics and Statistics, Uni¨ersity of Calgary, Calgary, Alberta, Canada T2N 1N4

View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector

M. F. Yousif ‡

Department of Mathematics, Ohio State Uni¨ersity, Lima, Ohio 45804

Communicated by Kent R. Fuller

Received May 20, 1996

A ring R is called right mininjective if every isomorphism between simple right ideals is given by left multiplication by an element of R. These rings are shown to be Morita invariant. If R is commutative it is shown that R is mininjective if and only if it has a squarefree socle, and that every image of R is mininjective if and only if R has a distributive lattice of ideals. If R is a semiperfect, right mininjective ring in which eR has nonzero right socle for each primitive idempotent e,itis shown that R admits a Nakayama permutation of its basic idempotents, and that its two socles are equal if every simple left ideal is an annihilator. This extends well known results on pseudo- and quasi-Frobenius rings. ᮊ 1997 Academic Press

A ring R is called quasi-Frobenius if it is left and right artinian and left and right selfinjective; equivalently if R has the ACC on right and left annihilators and is right or left selfinjective. Many other characterizations of these rings have been given but one question remains open: The Faith

* This research was supported by the NSERC under Grant A8075 and by The Ohio State University. Both authors acknowledge the Centre De Recerca Matematica,` Barcelona, Spain, and the Spanish Ministerio de Education y Ciencia for their support of some of the work done on this paper. † E-mail: [email protected]. ‡ E-mail: [email protected].

548

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. MININJECTIVE RINGS 549 conjecture asserts that every semiprimary right selfinjective ring is quasi- Frobenius. In this paper we study this question by examining rings with a very weak selfinjectivity hypothesis. Surprisingly, many of the basic proper- ties of pseudo- and quasi-Frobenius rings can be deduced in this general- ity. In particular, we give mild conditions which guarantee that the right and left socles of the ring are equal. Several applications are given which extend known sufficient conditions that a ring be quasi-Frobenius. Throughout this paper all rings have unity and all modules are unitary. The right and left annihilators of a subset X of a ring R are denoted rXŽ. and lXŽ., respectively, and we write J s JR Ž.for the Jacobson radical of R.If Mis a module we write ZMŽ.for the singular submodule of M, and we write socŽ.M for the socle of M. The right and left socles of R are Ž. Ž. denoted SrRlRs soc R and S s soc R . If P is a property of rings, a ring Rwhich is both a left and right P-ring will be referred to simply as a P-ring.

1. GENERALITIES

If R is a ring, a right module MR is called mininjecti¨e if, for each simple right ideal K of R, every R-morphism ␥ : K ª M extends to R; equivalently if ␥ s m и is left multiplication by some element m of M. Mininjective left modules are defined similarly. Clearly every injective module is mininjective. Our interest is in the right mininjecti¨e rings, that is, the rings R for which RR is mininjective. These rings were first introduced by Haradawx 15 , who studied the artinian case inwx 15 and wx 16 . We begin with several characterizations.

LEMMA 1.1. The following conditions are equi¨alent for a ring R:

Ž.1 R is right mininjecti¨e. Ž.2 If kR is simple, k g R, then lrŽ. k s Rk. Ž.3 If kR is simple and rŽ.Ž. k : ra,k,agR,then Ra : Rk. Ž.4 If kR is simple and ␥ : kR ª RisR-linear, then ␥ Ž.k g Rk.

Proof. GivenŽ. 1 , let 0 / a g lr Ž k .. Then ␥ : kR ª R is well defined by kr ¬ ar,so␥scи,cgR, byŽ. 1 . Then a s ck, proving Ž. 2 . The routine verification thatŽ. 2 « Ž.3 « Ž.4 « Ž.1 is omitted.

A ring R is called right principally injecti¨e if each R-morphism aR ª R, Ž a g R, extends to RRRª R . Clearly, every such ring and hence every right selfinjective ring. is right mininjective. 550 NICHOLSON AND YOUSIF

If all minimal right ideals of a ring R are summandsŽ for example, if R has zero right socle. then R is right mininjective. In particular, every semiprime ring is right and left mininjective. However, the ring R s FF,Fa field, is right and left nonsingular andŽ. so both socles are 0 F projective, but R is neither right nor left mininjective.

Remark 1.1. Every polynomial ring Rxwxis mininjective because both socles are zero.

n 1 Indeed, if K s kRwx x is simple where degŽ.k s n, then K s Kx q so n 1 kgkRwx x x q, a contradiction. Remark 1.2. The ring ޚ of integers is a commutative, noetherian, mininjective ring that is not artinian or principally injective.

Remark 1.3. If Sr is simple as a left ideal then R is right mininjective.

In fact, if kR is simple and rkŽ.:ra Ž.,k,agR,a/0, then rkŽ.sra Ž. Ž. because rk is maximal, so Ra is simple too. Hence Ra s Sr s Rk by hypothesis, and Lemma 1.1 applies.

Remark 1.4. A commutative local ring R is mininjective if and only if socŽ.R is simple or zero.

For if socŽ.R / 0, let K and M be simple ideals. Then K ( M because R is local, so M s cK for c g R because R is mininjective. Hence M s K and socŽ.R is simple. The converse is by Remark 1.3. Hence a commuta- tive artinian ring is quasi-Frobenius if and only if it is mininjective. We generalize this to the noncommutative caseŽ under weaker chain condi- tions. in Section 4.

Ł Remark 1.5. A direct product ig IiR of rings Riis right mininjective if and only if Ri is right mininjective for each i g I.

Remark 1.6. Rutterwx 22, Example 1 has an example of a two-sided artinian, right principally injective ring that is not left mininjective.

Remark 1.7. Camillowx 6 has an example of a commutative, local, 3 semiprimary, mininjective ring with J s 0 which is not artinian. The next result is half of the proof that mininjectivity is a Morita invariant.

2 PROPOSITION 1.2. If R is right mininjecti¨e, so is eRe for all e s e g R satisfying ReR s R. MININJECTIVE RINGS 551

Ž. Ž. Proof. Write S s eRe and let rkSS:ra, where k, a g S and kS is a simple right ideal of S. We claim first that kR is simple in R. For if kr / 0, r g R, then krReR / 0, so there exists b g R such that 0 / krbe s Ž.ke rbe g kS. Hence k g krbeS : krR, whence kR is simple. Thus it suf- Ž. Ž.Ž fices to show that rkRR:ra. Then a g Rk by Lemma 1.1 so a s . Ýn ea g Sk, as required. So let kr s 0, r g R, and write 1 s is1aebii, Ž.Ž. Ž. where aii, b g R. Then k erai e s kr ai e s 0 for each i so a erai e s 0 Ýn by hypothesis. Hence ar s is1 araii eb s 0, as required. The proof that right mininjectivity is inherited by matrix rings requires the following result.

LEMMA 1.3. Call a simple right ideal K of R ‘‘nice’’ if maps K ª RR can be extended to R ª R. If dK / 0 is ‘‘nice,’’ d g R, then K is ‘‘nice.’’ Proof. The map ␴ Ž.x s dx defines an isomorphism ␴ : K ª dK. Given y1 y1 ␥ : K ª RR we have ␥ (␴ : dK ª R so ␥ (␴ s c и , c g R, by hypothe- sis, and it follows that ␥ s Ž.cd и . Ž. PROPOSITION 1.4. A ring R is right mininjecti¨e if and only if Mn R is Ž. Ž. right mininjecti¨e for all some n G 1, where Mn R is the ring of all n = n matrices o¨er R. Ž. Proof. If S s MRn is right mininjective, so is R ( eSe11 11 by Proposi- Ž. tion 1.2, because Se11 S s S here eij denotes the matrix unit . Conversely, if R is right mininjective let kS be a simple right ideal of S. If row i of k is nonzero then ek1i /0, so, by Lemma 1.3, we may assume that k g eS11 . Again, if column j of k is nonzero then kej1 / 0sokS s kej1 S. Thus we may assume that k eSe, so write k k 0 in block form, k R. g 11 11 s 00 g Then kR is simple so Rk s lrŽ. k by Lemma 1.1. But then rkŽ. rk Ž.иии rk Ž. RRиии R rkSŽ.s .. ., .. . RRиии R whence

lrŽ. k 0 иии 0 Rk 0 иии 0 lrŽ. k 0 иии 0 Rk 0 иии 0 .. . lrSŽ. k sss...... Sk, ...... lrŽ. k 0 иии 0 Rk 0 иии 0 as required. 552 NICHOLSON AND YOUSIF

If M is the free right R-module of rank ␹ and we view end RŽ.M as the set of ␹ = ␹ column finite matrices over R, the proof of Proposition 1.4 goes through to give

COROLLARY 1.5. If R is right mininjecti¨e, so also is the endomorphism ring of any free right R-module. While principal injectivity is not a Morita invariantwx 20, Theorem 4.2 , combining Propositions 1.2 and 1.4 gives

THEOREM 1.6. Right mininjecti¨ity is a Morita in¨ariant. The next theorem gives another characterization of right mininjectivity based on the following ‘‘relative’’ version of the concept. If e is an idempotent in R, a right R-module MR is called eR-mininjecti¨e if each R-morphism ␥ : K ª M, where K : eR is a simple right ideal, extends to eR ª M, equivalently if ␥ s m и for some m g M. Clearly M is mininjec- tive if and only if it is R-mininjective.

LEMMA 1.7. If 1 s e1 q иии qeinRni,where the e are orthogonal idempotents, then a right module M is mininjecti¨e if and only if M is eRi -mininjecti¨e for each i.

Proof. Let ␥ : K ª MR be R-linear, where K is a simple right ideal. Ž. Then eKii/0 for some i so ␴ k s ekdefines an isomorphism ␴ : K ª y1 eKii.If Mis eR-mininjective we have ␥ (␴ s m и for some m g M, Ž. whence ␥ s mei и and M is mininjective. The converse is clear. LEMMA 1.8. Assume that eR ( fR, where e and f are idempotents. If a module MR is eR-mininjecti¨e then it is fR-mininjecti¨e. Proof. Let e s ab and f s ba, where a g eRf and b g fRe. Given ␥:XªM, where X : fR is simple, write K s Ä4k g eR N bk g X .We claim that K is a simple right ideal. For if 0 / k g K then bk / 0 because ksabkŽ.,so XsbkR. Thus, given kЈ g K, we have bkЈ s bkr, r g R,so kЈsabkЈ s abkr s kr g kR. This proves that K is simple. Ž. Ž . Now define ␥11: K ª M by ␥ k s ␥ bk . By hypothesis ␥1s m и for some m g M,so␥Ž.bk s mk s mabk Ž., and it follows that ␥ s Ž.ma и .

LEMMA 1.9. If e2 e in R and M M , then M is eR- s RiRs [igI mininjecti¨e if and only if each Mi is eR-mininjecti¨e.

Proof. Assume that each Mi is eR-mininjective. If ␥ : K ª M is R- linear, where K kR eR is simple, let ␥ Ž.K n M .If␲:M M s : : [ts1ttª t is the projection for each t, then ␲ tt(␥ s m и for some m ttg M by ␥ Ž. Ý XX␲␥Ž. hypothesis. If k s ig Iim then m ts tk smk tand it follows ␥ и Ýn that s m , where m s ts1 mt. MININJECTIVE RINGS 553

THEOREM 1.10. Let 1 s f1 q иии qfinRni,where the f are orthogonal Ä4Ä4 Ä 4 idempotents, and let e1,...,em : f1,..., fn be such that e1 R,...,eRm is a complete set of distinct representati¨es of the fi R. The following are equi¨alent: Ž.1 R is right mininjecti¨e. Ž. 2 eRiseRij-mininjecti¨e whene¨er 1 F i, j F m. Ž. 3 eRe is a right mininjecti¨e ring, where e s e1 q иии qem. Ž. Ž. Ž. Proof. 1 « 2 Given 1 , each eRi is mininjective by Lemma 1.9 with e s 1, so Lemma 1.7 asserts that eRikis fR-mininjective for each k. This provesŽ. 2 . Ž. Ž. Ž. 2«1 Fix i,1FiFm. Then 2 and Lemma 1.8 show that eRi is fRki-mininjective for all k s 1,...,n,so eR is mininjective as a right R-module by Lemma 1.7. This holds for each i so fRk is right mininjective for each k Ž.mininjectivity is preserved by isomorphisms . Finally, this shows that R n fRis mininjective by Lemma 1.9, provingŽ. 1 . Rks [ks1 Ž. Ž. 1m3 We have ReR s R by the choice of the ei,so Ris Morita equivalent to eRe and Theorem 1.6 applies.

We now show that a version of the C23 - and C -conditionsŽ seewx 18 . holds in a right mininjective ring.

PROPOSITION 1.11. Let R be a right mininjecti¨e ring. Ž. 22 MC2 If K ( eR is simple, e s e, then K s gR for some g s g. Ž. 22 MC3 If eR / fR are simple, e s e, f s f, then eR [ fR s gR for 2 some g s g. Ž. Proof. MC2 If ␥ : K ª eR is an isomorphism, let ␥ s c и , c g R. Ž. 2 Ž. Then cK s eR ­ JR so K / 0 and MC2 follows. Ž. Ž. Ž. MC3 We have eR [ fR s eR [ 1 y efR.If1yefRs0 we are Ž. Ž. 2 Ž. done. Otherwise, 1 y efR(fR so 1 y efRshR, h s h,byMC.2 Hence eh s 0so gseqhyhe is an idempotent such that eg s e s ge and hg s h s gh. It follows that eR [ fR s gR. Proposition 1.11 has a nice application to I-finite mininjective rings, where a ring R is called I-finite if it contains no infinite orthogonal family of idempotents.

THEOREM 1.12. Let R be I-finite and right mininjecti¨e. Then R ( R1 = R21, where R is semisimple and e¨ery simple right ideal of R2 is nilpotent.

Proof. Let 1 s e1 q иии qeni, where the e are orthogonal primitive idempotents with eRi simple if 1 F i F m and not simple if i ) m. 554 NICHOLSON AND YOUSIF

Claim. eReijs0seRe jifor all 1 F i F m - j F n. Ž Proof. If 0 / a g eReijthen a и : eRjªeR iis epic since eRiis sim- ple.Ž and so is an isomorphism since eRj is indecomposable . , a contradic- tion as eRjjis not simple. Now let 0 / b g eRei. Then b и : eRiªeRjis Ž. 2 monic since eRi is simple and so bR s fR where f s f by Proposition 1.11. Hence b и is epicŽ. since eRj is indecomposable , a contradiction as before. This proves the claim. Ž. Ž. If e s e1 q иии qem it follows that eR 1 y e s 0 s 1 y eRe. Hence e is a central idempotent and R1 s eR is semisimple. It remains to show that each simple right ideal K : Ž.1 y eRis nilpotent. If not, K s fR, 2 Ž. fsf.As K1ye/0, let Kejj/ 0 where j ) m. Thus fRe / 0, say 0/cgfRejj. Then c и : eRªfR is epic, a contradiction as before.

COROLLARY 1.13. Let R be right mininjecti¨e and I-finite with 1 s e1 q иии qeni, where the e are orthogonal, primiti¨e idempotents. Either some 2 eir R is simple or S s 0. We conclude this section with a fundamental fact about right mininjec- tive rings which will be used frequently below. If K and M are modules with K simple, write soc K Ž.M for the homogeneous component of M generated by K.

THEOREM 1.14. Let R be a right mininjecti¨e ring, and let k, m g R. Ž.1 If kR is a simple right ideal, then Rk is a simple left ideal. Ž.2 If kR ( mR are simple, then Rk ( Rm; in fact Rk s ŽRm . u for some u g R. Ž. Ž . 3 If kR is simple, soc kRR R s RkR is a simple ideal of R contained in soc RkŽ. R R . Ž. 4 Srl:S.

Proof. Ž.1If0/ak g Rk, define ␥ : kR ª akR by ␥ Ž.x s ax. Then ␥ 1 1 is an isomorphism so let ␥y s c и , c g R. Thus k s ␥y Ž.ak s cak g Rak, andŽ. 1 follows.

Ž.2If␴:kR ª mR is an isomorphism, write ␴ k s ␴ Žk .s mu, u g R. Clearly rkŽ.sr Ž␴k .so, as ␴ kR s mR is simple, Lemma 1.1 gives Rk s RŽ.Ž␴ k s Rm . u. Ž. Ž . Ž . 3 Write S s soc kRR R . Always RkR:S. Suppose ␴ : kR ª M:Ris an R-isomorphism. Then rkŽ.sr Ž␴k .so Rk s R␴ k by Lemma 1.1. Hence M s Ž.␴ kR: Ž.Rk R,soS:RkR. Thus S s RkR. Now let 0 / A : S, A an ideal of R.If M:Ais a simple right ideal then M ( kR. Hence if X is any right ideal isomorphic to kR, let ␥ : M ª X be MININJECTIVE RINGS 555 an R-isomorphism. Then ␥ s c и , c g R,so Xs␥Ž.M scM : cA : A. It follows that S : A, whence S s A and S is a simple ideal. Finally, Ž. Ž. Rk R : soc Rk R R always holds. Ž.4 This follows fromŽ. 1 .

If R is right mininjective, it follows fromŽ. 3 of Theorem 1.14 that every two-sided ideal of Sr is a direct sum of simple ideals. Moreover, if R is right and left mininjective and we write S s Srls S , the set of left homogeneous components of S is the same as the set of right homoge- neous components of S.

2. DUALITY CONSIDERATIONS

d Ž. If MR is a right R-module, define the dual of M as M s hom RRM, R . d d This is a left R-module, where, if r g R and ␭ g M , the map r␭ g M is defined by Ž.Ž.r␭ m s r␭ Ž.m for all m g M.

LEMMA 2.1. If M s mR is a principal module and T s rmŽ., then d M(lTŽ.slr Ž m .. Ž. Ž . Proof. If b g lT the map ␭bb: M ª R is well defined by ␭ mr s br. Ž. d Then b ¬ ␭b is an isomorphism lT ªM of left R-modules. The next result gives an important characterization of right mininjective rings in terms of duality. The equivalence of the first two conditions was observed by Bjork¨ wx 4, Proposition 3.1 for artinian rings.

PROPOSITION 2.2. The following are equi¨alent for a ring R:

Ž.1 R is right mininjecti¨e. Ž.2 Md is simple or zero for all simple right R-modules M. Ž.3 l Ž T . is simple or zero for all maximal right ideals T of R. Ž. Ž. d Proof. 1 « 2 Let ␥, ␦ g M , where MR is simple, and assume that 1 ␥/0. We have ␦ (␥y : ␥ Ž.M ª R so, since ␥ Ž.M is a simple right ideal 1 d of R, ␦ (␥y s a и byŽ. 1 , where a g R. Hence ␦ s a␥ in M , provingŽ. 2 . d Ž.2« Ž.3IfTis a maximal right ideal of R then lT Ž .( ŽRrT . by Lemma 2.1. Ž.3« Ž.1 Let ␥ : kR ª R be R-linear, where kR s K is a simple right ideal, and let ␫: K ª R be the inclusion map. If T s rkŽ., then d d d K (lTŽ.by Lemma 2.1, so K is simple byŽ. 3 . Thus ␥ s c␫ in K for some c g R,so␥scи, provingŽ. 1 . 556 NICHOLSON AND YOUSIF

In a quasi-Frobenius ring, the map T ¬ lTŽ.is an inclusion reversing bijection between the right ideals T and the left idealsŽ the inverse is L¬rLŽ... The next theorem identifies a condition under which a weak- ened form of this duality holds in a right mininjective ring. A ring R is called right Kasch if every simple right module embeds in R, equivalently if lTŽ./0 for every maximal right ideal T of R. If, in addition, R is right mininjective, lTŽ.is a simple left ideal by Proposition 2.2. In fact we have

THEOREM 2.3. Let R be a right mininjecti¨e ring which is right Kasch, and consider the map

␪ : T ¬ lTŽ. from the set of maximal right ideals T of R to the set of minimal left ideals of R. Ž.1 ␪is one-to-one. Ž.2 ␪is a bijection if and only if lrŽ. K s K for all minimal left ideals K 1 of R. In this case ␪y is gi¨en by K ¬ rKŽ.. Proof. Ž.1IfTis a maximal right ideal, then lT Ž ./0 by the Kasch hypothesis, so lTŽ.is simple by Proposition 2.2. Since T : rl Ž. T / R,we have T s rlŽ. T because T is maximal. Now Ž. 1 follows. Ž.2If␪is onto and K is a minimal left ideal, let K s lT Ž .,T:R. Then lrŽ. K s K follows. Conversely, assume that lr Ž. K s K for all mini- mal left ideals K. Claim.IfKis a minimal left ideal then rKŽ.is maximal.

Proof. Let rKŽ.:T, where T is maximal. Then K s lr Ž. K = lT Ž./0 by the right Kasch hypothesis, so K s lTŽ.because K is simple. Thus rKŽ.srl Ž. T = T, whence rK Ž.sT, proving the claim. By the claim we have a map ␸ given by K ¬ rKŽ., which we assert is the inverse of ␪. Indeed, ␸ (␪ carries T ¬ lTŽ.¬rl Ž. T s T by the calculation inŽ. 1 , while ␪ (␸ carries K ¬ rK Ž.¬lr Ž. K s K by hypothe- sis. This completes the proof ofŽ. 2 . Motivated by Theorem 2.3, we call a ring R a left minannihilator ring if every minimal left ideal K of R is an annihilator; equivalently, if lrŽ. K s K. Clearly, every semiprime ring is a minannihilator ring. Also, every right principally injective ring R is a left minannihilator ringŽ because R is right principally injective if and only if lrŽ. a s Ra for all a g R.Ž. Recall that by Lemma 1.1.Ž a ring R is right mininjective if and only if lr K .s K for every left ideal K s Rk for which kR is simple. Thus, for example, a commutative ring is mininjective if and only if it is a minannihilator ring. MININJECTIVE RINGS 557

The minannihilator condition and its relation to mininjectivity has been studied in artinian rings by Dieudonne´¨wx 9 , Storrer w 23 x , and Bjork wx 4 . The following result and its corollaries identify the close connection between right mininjective rings and left minannihilator ringsŽ see also Proposition 3.5. . The following notion is germaine: Call a ring R a right minsymmetric ring if kR is simple, k g R, implies that Rk is simple. These rings all have the property that Srl: S . Every right mininjective ring is right minsymmet- ric by Theorem 1.14, as is every commutative and every semiprime ring. As noted above, a right mininjective ring R which is left minsymmetric Ž Rk simple, k g R, implies kR simple. is a left minannihilator ring by Lemma 1.1. The converse requires right symmetry.

PROPOSITION 2.4. The following are equi¨alent for a left minannihilator ring R:

Ž.1 R is right mininjecti¨e. Ž.2 R is right minsymmetric. Ž. 3 Srl:S.

Proof. We haveŽ. 1 « Ž.2 by Theorem 1.14;Ž. 2 « Ž.3 always holds. Ž. Ž. Ž. Ž. 3«1 Given 3 , let kR be simple. Then k g Sl by 3 so let Rk = Rm, where Rm is simple. Thus rkŽ.:rm Ž.so rk Ž.srm Ž.because rkŽ.is maximal. Since R is a left minannihilator ring, Rk : lr Ž Rk .s lrŽ. Rm s Rm.As Rm is simple, it follows that Rk s lr Ž. Rk s lr Ž. k , provingŽ. 1 .

The proof thatŽ. 3 « Ž.1 in Proposition 2.4 also yields

COROLLARY 2.5. If R is a left minannihilator ring in which Sl is essential inR R, then R is right mininjecti¨e.

COROLLARY 2.6. A ring R is mininjecti¨e if and only if Srls S andRisa minannihilator ring

The following characterization of right minsymmetric rings has some independent interest as the second condition, without the restriction that kR be simple, characterizes the right principally injective ringsw 20, Lemma 1.1x .

PROPOSITION 2.7. The following are equi¨alent for a ring R: Ž.1 R is right minsymmetric. Ž.2 If kR is a simple right ideal, then lw kR l raŽ.xslkŽ.qRa for all a g R. 558 NICHOLSON AND YOUSIF

Proof. Ž.1 « Ž.2 Assume kR is simple and let a g R.If ak s 0 then kR l raŽ.skR and lk Ž.qRa s lk Ž., and Ž. 2 follows. If ak / 0 then lkŽ.qRa s R Žbecause lk Ž.is maximal byŽ.. 1 and kR l ra Ž.s0, and againŽ. 2 follows.

Ž.2« Ž.1IfkR is simple, let a f lkŽ.. Then kR l raŽ.s0so lkŽ.qRa s R byŽ. 2 . This shows that lkŽ.is maximal, provingŽ. 1 . A submodule N of a module M is said to be essential in M Žwritten ess N : M . if N l X / 0 for all submodules X / 0of M.

PROPOSITION 2.8. Suppose Srl: S in a ring R. Ž. ess Ž. Ž . 1 If SrR: R , then J R : ZRR. Ž. Ž.Ž. 2 If R is also semiperfect, then J R s ZRR . Ž. Ž. Proof. 1 Since Srl: S we have JSr: JS ls 0, so S r: rJ. Because ess Ž. ess Ž. SrR:R, this gives ra: RRfor all a g J, that is, J : ZRR. Ž. Ž.Ž. 2 We have ZRR :JR in any semiperfect ring because any Ž.one-sided ideal A ­ JR Ž.contains a nonzero idempotent.

Since right mininjective rings R have Srl: S by Theorem 1.14, the following result is immediate.

THEOREM 2.9. Assume that a ring R is semiperfect and right mininjecti¨e ess Ž. Ž . and satisfies SrR: R . Then J R s ZRR. A module is said to have squarefree socle if every nonzero homogeneous component of its socle is simple. For a ring R, Sr is squarefree if and only if ␥ Ž.k g kR for all R-linear maps ␥ : kR ª R with kR simple.Ž Note that Ž. . insisting instead that ␥ k g Rk characterizes right mininjectivity. If Sr is squarefree it is clear that every minimal right ideal is two-sided; the next result provides a partial converse.

THEOREM 2.10. Assume that kR s Rk whene¨er kR is a simple right ideal of the ring RŽ. for example, if left and right ideals are two-sided . Then R is right mininjecti¨e if and only if Sr is squarefree. Proof. Let kR be simple, k g R.If Ris right mininjective it suffices by Theorem 1.14 to show that RkR s kR. But this follows because kR is two-sided. Conversely, let 0 / ␥ : kR ª R be R-linear. Then ␥ Ž.kRskR Ž. because Sr is squarefree, whence ␥ k g kR : Rk and R is right mininjec- tive by Lemma 1.1.

COROLLARY 2.11. A commutati¨e ring is mininjecti¨e if and only if it has squarefree socle. MININJECTIVE RINGS 559

A module M is called distributi¨e if A l Ž.Ž.B q C s A l B q Ž.AlCholds for all submodules A, B, and C of M. These modules have been studied by Camillowx 5 and their relationship to mininjectivity is given in the following theorem. Call a ring R right duoŽ. left duo if every right Ž.left ideal of R is two-sided.

THEOREM 2.12. The following conditionsŽ. and their left-right analogues are equi¨alent for a duo ring R: Ž.1 RrA is right mininjecti¨e for all ideals A of R. Ž. 2 RR is distributi¨e. Proof. Observe that RrA inherits the duo hypothesis. Hence, ifŽ. 1 Ž. holds, then RrA R r A has squarefree right socle by Theorem 2.10. Thus Ž. Ž. RrAR has squarefree socle, and 2 follows by a theorem of Camillow 5, Ž. Ž.Ž. Theorem 1x . Conversely, 2 implies that RrA RRs RrA rAhas square- free socle by Camillo’s theorem, soŽ. 1 follows from Theorem 2.10.

We conclude this section with some conditions which imply that Sr is right finite dimensional in a right mininjective ring R.

LEMMA 2.13. Let kR and mR denote minimal right ideals in a right minsymmetric ring R. The following conditions are equi¨alent: Ž.1 If kR [ mR is direct then Rk [ Rm is also direct. Ž.2 If Rk s Rm then kR s mR. Proof. Ž.1 « Ž.2 This holds in any ring because kR and mR are simple.

Ž.2« Ž.1 This holds because Rk and Rm are simple ŽR is right minsymmetric. .

PROPOSITION 2.14. Let R be a right mininjecti¨e ring and assume that R has n distinct maximal right ideals. If either of the following conditions holds then the right Goldie dimension of Sr is at most n. Ž.1 R is a right Kasch, left minannihilator ring. Ž.2 R satisfies the conditions in Lemma 2.13. Ž. иии Proof. 1 Suppose K1 [ [ K nq1 is a direct sum of simple right ideals. Then rKŽ.i is a maximal right ideal for each i by Theorem 2.3, and Ž. Ž. Ž. Ž. rKijsrK implies K iis lr K s lr K jjs K , whence i s j. Ž. иии 2 Suppose kR1 [ [ kRnq1 is a direct sum of simple right Ž. Ž. Ž. ideals. Each rkiiis a maximal right ideal, and rk srkjimplies Rkijs Rk by right mininjectivity. Hence kRiskR jby the condition in Lemma 2.13, so i s j. 560 NICHOLSON AND YOUSIF

The next result obtains the same conclusion but with a restriction on the number of maximal left ideals.

PROPOSITION 2.15. Let R be a mininjecti¨e ring. If R has n distinct maximal left ideals, then the right Goldie dimension of Sr is at most n. Proof. We begin with a result of independent interest.

Claim.IfkR and mR are minimal right ideals of R with kR l mR s 0, then lkŽ.qlm Ž .sR. Proof. By right mininjectivity, Rk and Rm are minimal left ideals, so lkŽ.and lm Ž .are maximal left ideals. Hence we must show that lk Ž./ lmŽ.. But lk Ž.slm Ž.implies kR s mR by left mininjectivity, a contradic- tion. This proves the claim.

Now let M1,...,Mn be the maximal left ideals. Suppose kR1 [иии [ Ž. kRnq1 is a direct sum of minimal right ideals. Then lk1 is maximal by Ž. Ž .Ž. Theorem 1.14, say lk11sM after relabeling the Mi. Now lk21/M Ž.Ž. by the claim so after relabeling let lk22sM. We continue this process Ž. Ž.Ž. to get lkiisM for 1 F i F n. But then lknq1sMttslk for some ts1,...,n, contrary to the claim.

3. SEMIPERFECT MININJECTIVE RINGS

A ring R is quasi-Frobenius if it is artinian and admits a ‘‘Nakayama’’ permutation of its basic set Ä4e12, e ,...,enof primitive idempotents, that is, Ä4 Ž. a permutation ␴ of 1, 2, . . . , n such that soc Rei ( Re␴ irJe␴ i and Ž. soc eR␴ iii(eRreJfor each i. It is well known that any ring with perfect dualityŽ. that is, any pseudo-Frobenius ring admits such a permutation. If e and f are primitive idempotents in a semiperfect ring R, then Ž.eR, Rf is called an i-pairŽ.Ž.Ž. injective pair if soc eR ( fRrfJ and soc Rf ( RerJe. These Nakayama permutations are useful in showing that right selfinjectiv- ity implies left selfinjectivity because of a result of Fullerwx 11 : If R is left artinian and e is a local idempotent of R, then eR is injective if and only if there exists a local idempotent f in R such that Ž.eR, Rf is an i-pair; in this case Rf is also injective. This theorem was investigated by Baba and Oshirowx 2 in the semiprimary case. Inwx 20 we showed that a more general class of ringsŽ the generalized pseudo-Frobenius rings. admits Nakayama permutations. In this section we are interested in the weakest conditions which will guarantee the existence of such a permutation. More precisely we show that if R is a semiperfect, right mininjective ring for which socŽ.eR / 0 for every local idempotent e of R, then R admits a Nakayama permutation. MININJECTIVE RINGS 561

An idempotent e in a ring R is called local if the ring eRe is a local ring; equivalently if eJ is the unique maximal right ideal in eR. It is well known that a ring is semiperfect if and only if 1 can be written as a sum of orthogonal, local idempotents.

LEMMA 3.1. Consider the following conditions on a ring R:

Ž.1 R is right mininjecti¨e. Ž. 2 Sr e is simple or zero for each local idempotent e g R.

Then Ž.1 « Ž.2; and Ž.2 « Ž.1 if e¨ery simple right ideal K satisfies Ke / 0 for some local idempotent e g R.

Proof. Ž.1 « Ž.2 It is well knownwx 8, Lemma 58.4 that lJeŽ.( Ž.d 2 Ž. Ž .d Ž. eRreJ for any e s e g R. Hence Ser :lJe(eRreJ and 2 follows from Proposition 2.2.

2 Ž.2« Ž.1IfKis a simple right ideal and Ke / 0, where e s e is Ž. local, let k s ke / 0, k g K. Then k g Serrso Rk s Se by 2 . Now Ž. Ž. suppose 0 / ␥ : K ª RR is an R-morphism. Then ␥ kR(Kso ␥ k s Ž. Ž. ␥kegSer sRk. Hence 1 follows from Lemma 1.1. Since 1 is a sum of local idempotents in a semiperfect ring, this gives the following important characterization of semiperfect right mininjective rings.

THEOREM 3.2. Let R be a semiperfect ring. Then R is right mininjecti¨eif and only if Sr e is simple or zero for each local idempotent e g R. We now turn to the structure of an arbitrary semiperfect right mininjec- tive ring R. The first result includes some important criteria that the two socles be equal.

PROPOSITION 3.3. Let R be a semiperfect, right mininjecti¨e ring.

Ž.1 Sr is semisimple and artinian as a left R-module.

Ž.2 R is right Kasch if and only if Sr e / 0 for each local idempotent e g R. 2 Ž.3 If 0 / k g soc ŽeR ., where e s e is local, then Rk is simple. Ž.4 If R is right Kasch, the following conditions are equi¨alent: Ž. a SrlsS. 2 Ž.b lr Ž K .s K for e¨ery simple left ideal K : Re with e s e g R local. Ž. Ž . 2 c soc Re s Sr e for e¨ery local e s e g R. 2 Ž.d soc ŽRe . is simple for e¨ery local e s e g R. 562 NICHOLSON AND YOUSIF

Ýn Proof. Write 1 s is1 eii, where the e are local idempotents in R. Ž. Ýn Then 1 follows because Sris s1Seri.If M Ris a simple module, let Ž. Me / 0 for a local idempotent e. Then M ( eRreJ. We have Srs lJ d Ž.dŽ. because RrJ is semisimple, so M ( eRreJ ( lJesSer . This proves Ž. Ž.d half of 2 ; the converse is true because Ser ( eRreJ . Finally, 0 / k g Ž. Ž. Ž . Ž . soc eR : Srl: S implies that lk =JqR1ye. But J q R 1 y e is maximal if e is local, andŽ. 3 follows.

Ž.a« Ž.b Let K : Re be a simple left ideal, where e is local. We have KJ s 0byaŽ.Ž.Ž. sorK =Jq1yeR.As Jq Ž.1yeRis maximal, Ž. Ž . Ž. Ž. rK sJq1yeR,so K:lr K s lJ lRe s Srrl Re s Se. Thus KslrŽ. K by Lemma 3.1. Ž. Ž. Ž. b«c Note that Ser is simple by 2 and Theorem 3.2. Let K:Re be simple. Then rKŽ.= Ž1yeR .so rK Ž.:Jq Ž1yeR . be- cause J q Ž.1 y eRis the unique maximal right ideal containing Ž. 1 y eR. But thenŽ. b gives

K s lrŽ. K = lJq Ž1yeR .slJ Ž.lRe s Ser /0.

Ž. Since K is simple, K s Ser and c follows. Ž.c« Ž.d This follows byŽ. 2 and Theorem 3.2. Ž. Ž. Ž. Ž . 2 d«a Given d we have soc Re s Ser for each local e s e by Ž. 2 . Let 1 s e1 q иии qeni, where the e are orthogonal local idempotents. Then

nn

SlirssocŽ.Re s Sei:Sr. [[is1is1

Hence Srls S by Theorem 1.14. The next result will be used repeatedly below.

LEMMA 3.4. Let e and f be local idempotents in a right mininjecti¨ering R.If eR and fR contain isomorphic simple right ideals, then eR ( fR. Proof. Let ␣: K ª fR be monic, where K : eR is a simple right ideal. By hypothesis ␣ s a и for some a g R, and we may assume that a g fRe. Ž Hence a и : eR ª fR is R-linear. We have Srl: S because R is right .Ž. mininjective so 0 / ␣ K s aK : aSrl: aS . This shows that a f J,so aeR s aR ­ fJ. Hence a и is onto fR because f is local. But then a и is one-to-one because fR is projective and eR is indecomposable. As the first application of Lemma 3.4 we can characterize the commuta- tive, semiperfect, mininjective rings. MININJECTIVE RINGS 563

PROPOSITION 3.5. The following are equi¨alent for a commutati¨e, semiperfect ring R:

Ž.1 R is mininjecti¨e. Ž.2 soc ŽRe . is simple or zero for all local idempotents e. Ž.3 R is a finite product of local rings whose socles are simple or zero. Ž.4 soc ŽR . is squarefree.

Proof. Ž.1 « Ž.2 If soc ŽRe ./ 0 let K ( M, where K, M : Re are simple. Then M s cK for c g R by mininjectivity, so M s K. This proves Ž.2.

Ž. Ž. 2«3 This follows because 1 s e1 q иии qemiin R, where the e are orthogonal, local idempotents. Ž.3 Ž.4 We have socŽ.R m socŽ.Re , where the e are as « s [is1ii Ž. Ž. above, and soc Reij` soc Re when i / j by Lemma 3.4. Ž.4« Ž.1 This follows from Corollary 2.11.

Recall that a ring R is called basic if 1 s e1 q иии qeni, where the e are orthogonal local idempotents and Reij( Re implies i s j. In this case we have:

THEOREM 3.6. Let R be a basic, semiperfect, right mininjecti¨e ring. Then:

Ž. 1 Let 1 s e1 q иии qeni, where the e are orthogonal local idempo- Ž. tents. If K is any simple right ideal of R then K : ei R for some unique i s 1,...,n.

Ž.2 soc ŽeR . is either 0 or a homogeneous component of Sr for all local 2 e s e g R. Ž. Proof. 1IfKskR is simple, we have k s ek1 qиии qekni.If ek/ 0/ekjijifor i / j, then ekR(kR ( ekR, whence eR(eRjby Lemma Ž. 3.4, a contradiction. Hence k s eki for some i, and 1 follows. Ž. Ž. 2 We may assume that e s e11, where e ,...,enare as in 1 . If Ž. soc eR / 0, let K : eR be simple. If KЈ ( K, KЈ : R then KЈ : eRi for some i. Hence i s 1 by Lemma 3.4, andŽ. 2 follows. We can now prove our main theorem, which obtains many properties of pseudo-Frobenius rings, but with much weaker hypotheses. For conve- nience, call a ring R right minfull if it is semiperfect, right mininjective and socŽ.eR / 0 for each local idempotent e g R. 564 NICHOLSON AND YOUSIF

THEOREM 3.7. Let R be a right minfull ring. Then: Ž.1 R is right and left Kasch. 2 Ž.2 soc ŽeR . is homogeneous for each local e s e g R. Ž. 2 3 Sr e is a simple left ideal for each local e s e g R. Ž.4 The following conditions are equi¨alent: Ž. a SrlsS. 2 Ž.b lr Ž K .s K for e¨ery simple left ideal K : Re, where e s e g R is local. Ž. Ž . 2 c soc Re s Sr e for all local e s e g R. 2 Ž.d soc ŽRe . is simple for all local e s e g R.

Furthermore, if e1,...,en are basic, orthogonal, local idempotents, there exist elements k1,...,kn in R and aŽ. Nakayama permutation ␴ of Ä41,...,n such that the following hold for each i s 1,...,n: Ž. 5 kRii:e R and Rk i: Re␴i. Ž. 6 kRi (eR␴ i re␴ iiii J and Rk ( Re rJe . Ž. 7 Rkirs Se␴i.

Ž.8Ä4Ä4kR1 ,...,kn R and Rk1,...,Rkn are complete sets of distinct representati¨es of the simple right and left R-modules, respecti¨ely. Proof. We begin withŽ.Ž. 5 ᎐ 8 . For each i s 1,...,n choose a simple right ideal K ii: eR.As Ris semiperfect, K i( eR␴ireJ␴ifor some ␴igÄ41,...,n. This map ␴ is a permutation ofÄ4 1, . . . , n because ␴ i s ␴ j Ž.Ž implies that K ij( K , whence eR i(eR jby Lemma 3.4 and i s j be- . cause the ei are basic . If ␥ : eR␴ i reJ␴ iiªKis an isomorphism, write Ž. ŽŽ.. ki s␥e␴ i qeJ␴ iiii. Then kRsK so k g eRei␴iproving 5 and kRi (eR␴ i reJ␴ ii. Because k g Sr: Sl, we obtain

lkŽ.ii=JqR Ž1ye .sJeii[ R Ž1 y e ..

ŽŽ.. Ž. But Rr Jeiii[ R 1 y e ( Re rJeiis simple, so it follows that lkis Ž. Ž. Jeii[ R 1 y e and hence that Rkii( Re rJei. This proves 6 . Now ob- Ž. serve that kiis ke␴irgSe␴ir. Hence 7 follows because Se␴iis simple Ž.by Theorem 3.2 . Since R is semiperfect it has exactly n isomorphism classes of simple rightŽ or left . modules, so Ž. 6 implies both Ž. 8 and Ž. 1 . To prove Ž. 2 , let Ž. K:eRijbe simple. Then K ( kRfor some j by 6 , so j s i by Lemma 3.4. Hence socŽ.eRi is homogeneous. NowŽ. 2 follows because each local idempotent e can be included in a basic set of local idempotents. Similarly, Ž.3 follows from Ž. 7 . MININJECTIVE RINGS 565

It remains to proveŽ. 4 : We have Ž. a « Ž.b « Ž.c « Ž.a by Proposition Ž. Ž. Ž. 3.3, and c m d because Ser is simple by 3 . COROLLARY 3.8. If R is a right minfull, left minannihilator ring then

Slrs S is finite dimensional as a left R-module. COROLLARY 3.9. A right minfull ring is left minfull if and only if it is left mininjecti¨e. COROLLARY 3.10. Let R be a minfull ring. Then: Ž. 1 SrlsSsS. Ž.2 soc ŽeR .s eS and socŽ.Re s Se are simple for all local idempotents e. Ž.3 S is right and left finite dimensional. Ž.4 R is a Kasch ring. Ž.5 R is a minannihilator ring.

Ž.6 If e1,...,en is a basic set of local idempotents, there exist elements k1,...,kinRn ,and a permutation ␴ of Ä41,...,n such that the following hold for all i s 1,...,n: Ž. Ž . Ž . a Rkis soc Re␴iiii( Re rJe and k R s soc eR i(eR␴ireJ␴i.

Ž.bÄ4Ä4kR1 ,...,kn R and Rk1,...,Rkn are sets of distinct represen- tati¨es of the simple right and left R-modules, respecti¨ely. Proof. Since R is right and left mininjective,Ž. 5 follows by Corollary 2.6.

COROLLARY 3.11. If R is a semiperfect, right mininjecti¨e ring, the following are equi¨alent: Ž.1 R is right minfull. Ž.2 R is right Kasch and soc ŽeR . is homogeneous for each local 2 e s e g R. Proof. ConditionŽ. 1 implies Ž. 2 by Theorem 3.7. Assume Ž. 2 . If K is a 2 simple right ideal of R, then eK / 0 for some local e s e g R so Ž. K(eK : eR. Thus, if e1,...,en are basic idempotents, it follows Kasch Ž. that every simple right R-module embeds in eRikfor some i. If soc eRs0 for some k, then some eRi with i / k contains two nonisomorphic simple right ideals, contrary toŽ. 2 . This provesŽ. 1 . Hence Theorem 3.6 shows that a basic, semiperfect, right mininjective ring is right minfull if it is right Kasch. In particular, if R is any basic, semiperfect, right minfull ring, we can improve uponŽ. 2 and Ž. 4 of Theorem 3.7, and obtain anŽ. improved left version of Theorem 3.6Ž. 1 . 566 NICHOLSON AND YOUSIF

PROPOSITION 3.12. Let R be a basic right minfull ring and let 1 s e1 q иии qeni, where the e are orthogonal local idempotents. Ž. 1 If k1,...,kn in R are chosen as in Theorem 3.7, we ha¨e

socŽ.eR soc Ž.R Rk R iks iRRis is a simple ideal. Ž. 2 SrlsS if and only if R is a left minannihilator ring; and in this case Ž. the only simple left ideals of R are the Sri e s soc Rei for i s 1, . . . , n. Ž. Proof. 1 Since eRij`eR when i / j, this follows from Theorems 1.14 and 3.6 because each simple right ideal is contained in eRi for some i s 1,...,n. Ž. 2 Assume that Srls S .If K is a simple left ideal, then Kei/ 0 for some i so, as Keiliri: SesSe, we have Keiris Se by Lemma 3.1. If Kejr/ 0 for j / i, then SejsKej( Keis Seri, contrary to Theorem 3.7. Ž. Ž . It follows that K s Keiris Se. Hence rK =Jq1yeRi, and so

lrŽ. K : lJ Ž.lReirs S l Re iris SesK.

Ž. Thus lr K s K, so we have proved that Srls S implies that R is a left minannihilator ring. As the converse holds by Proposition 3.3, this proves Ž.2. Turning toŽ. left and right minfull rings, we obtain the following theorem.

THEOREM 3.13. The following are equi¨alent for a ring R: Ž.1 R is minfull Ž.2 R is semiperfect and the dual of e¨ery simple R-module is simple.

Ž.3 R is semiperfect and Srl e and eS are both simple for all local 2 esegR. Ž. Ž. Proof. 1 « 2 This follows by Corollary 3.10 because, if MR is 2 d d simple and Me / 0 where e s e is local, then M ( Ž.eRreJ ( Ž. Ž. lJesSer ssoc Re is simple. Ž.2« Ž.1 Given Ž. 2 , R is mininjective by Proposition 2.2, whence 2 Ž. Ž.d SrlsS.If e seis local, this gives soc Re s SelsSe r( eRreJ / 0, and R is left minfull. Similarly, R is right minfull. Ž. Ž. Ž .d Ž.d 2m3 This follows because Serl( eRreJ and eS ( RerJe 2 for all local e s e. MININJECTIVE RINGS 567

Theorem 3.13 is similar in spirit to a well known result about pseudo- Frobenius rings. Here a ring R is called right pseudo-FrobeniusŽ. PF-ring if every faithful right module is a generator, equivalently if R is a semiperfect, right selfinjective ring with essential right soclewx 21 . Theo- rem 3.7 highlights the following class of rings: Call a ring R right min-PF ess if R is a semiperfect, right mininjective ring in which SrR: R and 2 lrŽ. K s K for every simple left ideal K : Re, where e s e is local. Note that it is well known that right PF-rings are right and left min-PF.

THEOREM 3.14. If R is a right min-PF ring, then: Ž.1 R is right and left Kasch. Ž. Ž . 2 JsZRR . Ž. 3 SrlsSsS is left finite dimensional.

Ž.4 If e1,...,en is a basic set of local idempotents, there exist elements k1,...,kinRn ,and a permutation ␴ of Ä41,...,n such that the following hold for all i s 1,...,n: Ž. a kRii:e R and Rk i: Re␴i. Ž. b kRi (eR␴ i re␴ iiii J and Rk ( Re rJe .

Ž.cÄ4Ä4kR1 ,...,kn R and Rk1,...,Rkn are complete sets of distinct representati¨es of the simple right and left R-modules, respecti¨ely. Ž. Ž . d soc Re␴ iis Rk s Se␴iii( Re rJe is simple for each i.

Ž.e soc ŽeRi ./0 is homogeneous with each simple submodule iso- morphic to e␴i RreJ␴i. Proof. ConditionŽ. 1 and Ž. a through Ž. d of Ž. 4 all follow from Theorem 3.7,Ž. 2 is by Theorem 2.9, and Ž. 3 is by Corollary 3.8. To prove Ž. e , let Ž. K:eRijbe simple. Then K ( kR:eRjfor some j by c , so j s i by Lemma 3.4. This provesŽ. e .

COROLLARY 3.15. Suppose R is a semiperfect, left minannihilator ring in ess ess Ž which SrR: R and S lR: R for example, if R is a semiprimary, left minannihilator ring.. Then: Ž.1 R is a right min-PF ring which is left finite dimensional. Ž.2 If k g R, Rk is simple if and only if kR is simple. Ž. Ž . Ž . Ž . 3 JRsZRRRsZR . Proof. Ž.1 By Corollary 2.5, R is right mininjective. Hence R is right min-PF; it is left finite dimensional by Theorem 3.14 because this holds for Sl. Ž.2IfkR is simple then Rk is simple by Theorem 1.14. If Rk is simple, let rkŽ.:Twhere T is a maximal right ideal. Then Rk s lrŽ. k = lTŽ./0 by right Kasch. Hence Rk s lTŽ.,soT:rl Ž. T s rk Ž..AsT is maximal, T s rkŽ., whence kR is simple. 568 NICHOLSON AND YOUSIF

Ž. Ž. Ž. 3 We have J s ZRRRby Theorem 2.9, and ZR:Jbecause R ess Ž. is semiperfect. But Srls S : RR, and it follows that J : ZRR.

THEOREM 3.16. Suppose R is a right min-PF ring. If P and Q are projecti¨e left R-modules with soc P ( soc Q, then P ( Q.

Proof. Let Ä4e1,...,en be a basic set of orthogonal idempotents. If P and Q are indecomposable then P ( Reij, Q ( Re where soc Re i( soc Rej. By Theorem 3.14, there exists a permutation ␶ ofÄ4 1, 2, . . . , n such Ž. Ž. that soc Rei ( Re␶ irJe␶ ijand soc Re ( Re␶jrJe␶j. Hence i s j, and P ( Q. In general, P Re and Q Re . Since soc P soc Q we ( [igIjij( [gJ ( have socŽ.Re soc Ž.Re . Since each socŽ.Re is simple, it [igIjij( [gJ i follows from the Krull᎐Schmidt᎐Azumaya theorem that there is a bijec- Ž. Ž . tion ␪: I ª J such that soc Rei ( soc Re␪ i for each i g I. It follows from the indecomposable case that P ( Q. We conclude with some results on Morita invariance, and the following resultsŽ. of interest in their own right will be needed. Recall that a ring R is called semilocal if RrJ is semisimple artinian.

LEMMA 3.17. If R is a ring, each of the following is a Morita in¨ariant property. Ž. ess 1 SrR: R . Ž. 2 R is semilocal and Srl: S . Ž. Ž.Ž. Proof. Write R s MRnn, where n G 1. Then MSr:soc RRbe- cause kR being simple, k g R, implies keij R is simple for every matrix Ž. Ž . unit eij. Moreover, MSn r ssoc RR when R is semilocal because Sr s lJŽ.in that case. It follows that propertyŽ. 2 passes from R to R; the same is true of propertyŽ. 1 because every right ideal of R consists of all n matrices whose columns come from some submodule T : R . 2 Now let Q s eRe, where e s e g R satisfies ReR s R.If kQ is simple, k g Q, then kR is simple as in the proof of Proposition 1.2. It follows that Ž. Ž. soc QQr: eS e; the other inclusion is proved similarly. Thus soc QQs eSr e and propertiesŽ. 1 and Ž. 2 pass from R to Q Žthe semilocal hypothesis is not needed. .

THEOREM 3.18. Each of the following classes of rings is Morita in¨ariant. Ž.1 The right minfull rings. Ž.2 The right minfull rings R in which soc ŽRe . is simple for each local 2 e s e g R. Ž.3 The right min-PF rings. MININJECTIVE RINGS 569

Proof. Ž.1 A semiperfect right mininjective ring R is right minfull if and only if, for any projective module PR / 0, there exists an exact Ž. sequence 0 ª S ª P, where SR / 0 is semisimple. Thus 1 follows from Theorem 1.6. Ž.2 By Theorem 3.7Ž. 4 , a right minfull ring R satisfies the condition Ž. Ž. Ž. in 2 if and only if Srls S . Thus 2 follows from 1 and Lemma 3.17. Ž. Ž. 2 3 Let Q denote either MRn , where n G 1, or eRe, where e s egRand ReR s R.If Ris a right min-PF ring thenŽ. 1 , Theorem 3.7, Ž. and Lemma 3.17 show that Q is a right minfull ring in which soc QQ s Ž.ess soc QQQ : Q . Hence Q is right min-PF, again by Theorem 3.7.

4. APPLICATIONS

D-Rings

A ring R is called a D-ring if rlŽ. I s I and lr Ž L .s L for every right ideal I and left ideal L of R. Every D-ring is semiperfect, and D-rings with ACC on leftŽ. or right annihilators are easily seen to be quasi- Frobenius. These D-rings have been investigated by several authorsŽ see Hajarnavis and Nortonwx 13 for information. but no characterization is available in the literature. The following result provides a characterization. Haradawx 15 calls a ring R right simple-injecti¨e if every R-morphism with simple image from a right ideal of R to R is given by left multiplica- tion by an element of R. Such rings are clearly right mininjective.

THEOREM 4.1. A ring R is a D-ring if and only if R is a semiperfect, simple-injecti¨e ring with socŽ.eR / 0 for e¨ery local idempotent e of R. The proof of Theorem 4.1 depends on the following results, which are of independent interest. LEMMA 4.2. Let R be a right Kasch, right simple-injecti¨e ring. Then Ž.1 rl Ž I .s I for e¨ery right ideal I of R. Ž. 2 SrlsS. Proof. If I is a right ideal of R, and b g rlŽ. I , b f I, let MrI be a maximal submodule of Ž.bR q I rI. Since R is right Kasch, let Ž. Ž. ␦:bR [ I rM ª RR be an embedding. If ␥ : bR q I ª R is defined by ␥Ž.xs␦ ŽxqM ., then ␥ s c и , c g R,socI s ␥ Ž.I s 0. This gives cb s 0 because b g rlŽ. I . But cb s ␦ Žb q M ./ 0, and we have proved that rlŽ. I : I. Hence rl Ž. I s I. In particular rlŽ. a s aR for all a g R,so Ris left principally injective, and hence left mininjective. As R is also right mininjective, Srls S by Theorem 1.14. 570 NICHOLSON AND YOUSIF

PROPOSITION 4.3. Assume that R is a semiperfect, right simple-injecti¨e ring in which socŽ.eR / 0 for e¨ery local idempotent e of R. Then: Ž.1 rl Ž I .s I for e¨ery right ideal I of R. Ž. 2 SrlsSsS is essential in RRR and in R. Ž.3 soc ŽeR .s eS and socŽ.Re s Se are simple for e¨ery local idempo- tent e g R.

Ž.4If e1,...,en are basic local idempotents thenÄ4 e1 S,...,en S and Ä4 Se1,...,Sen are systems of distinct representati¨es of the simple right and left R-modules, respecti¨ely. Ž.5 R is right and left finite dimensional. Ž. Ž . Ž . Ž . 6 ZRRRsZR sJR. Proof. As R is right minfull, it is a right Kasch ring by Theorem 3.7. Ž. Ž . Hence Lemma 4.2 gives 1 and the fact that Srls S . Thus soc Re s Se is simple for each local idempotent e, again by Theorem 3.7. Since R is left principally injective byŽ. 1 , this shows that R is left minfull. So Theorem 3.7 implies that socŽ.eR s eS is simple for each local idempotent e, provingŽ. 3 . Now Ž. 4 follows from Ž. 7 and Ž. 8 of Theorem 3.7. ess Ž. We now show that SrR: R . Choose e1,...,enas in 4 and let Ž.Ž. Ž. 0/tgeRii. Then R 1 y e : lt so, as J q R 1 y eiis the unique Ž. Ž.Ž. maximal left ideal of R containing R 1 y eii, we have lt :JqR1ye . Ž. Ž . Thus rl t = rJleRiiseS,sotR = eS ibecause R is left principally Ž.ess injective. As t / 0 was arbitrary in eRii, this shows that soc eR : eRi. ess It follows that S s SrR: R . Finally, let 0 / b g R, and let ␣: bR ª RR have simple image. Then ␣saиfor some a g R so ab s ␣Ž.b / 0 while abJ : ␣Ž.bR J s 0. Thus Ž. ess 0 / ab g Rb l lJ sRb l S. This shows that S : R R and so proves Ž.2 . But then Ž. 5 follows fromŽ. 2 and Ž. 3 , and Ž. 6 follows from Theorem 2.9. Proof of Theorem 4.1. The necessity of the condition follows fromw 13, Lemma 3.2, Theorem 3.9, Proposition 5.2x . Conversely, since R is right simple-injective, Proposition 4.3 implies that socŽ.Re / 0 for all local idempotents e. Hence R is a D-ring byŽ. the right and left versions of Lemma 4.2.

Annihilator Chain Conditions We begin with a useful condition that a ring is semiprimary.

LEMMA 4.4. Suppose R is a semiperfect, right mininjecti¨e ring with ess SrR: R . If either R or RrSr has ACC on right annihilators, then R is semiprimary. MININJECTIVE RINGS 571

Ž. Ž . Ž. Proof. We have JRsZRR by Theorem 2.9. Thus JR is nilpotent if R has ACC on right annihilators. Now suppose R s RrSr has ACC on right annihilators. A straightforward application of Lemma 2.1 ofwx 20 shows that JRŽ.is right T-nilpotent. Thus JRŽ.is nilpotent, whence JRŽ. is nilpotent.

PROPOSITION 4.5. Let R be a right min-PF ring with ACC on right annihilators. Then R is left artinian.

Proof. Such a ring R is semiprimary by Lemma 4.4, so Srls S is left finite dimensional by Theorem 3.14. Thus R is left artinian byw 7, Lemma 6.x The following characterization of quasi-Frobenus rings will be used repeatedly. For a proof seewx 17, p. 342 .

LEMMA 4.6. An artinian ring R is quasi-Frobenius if and only if Srls S and both socŽ.Re and soc Ž.eR are simple for e¨ery primiti¨e idempotent e g R.

PROPOSITION 4.7. Suppose that R is a min-PF ring. If either R has ACC on right annihilators or RrsocŽ.R is right Goldie, then R is quasi-Frobenius. ŽŽ.. Proof. By Theorem 3.13, Srls S which we write as soc R and both 2 socŽ.Re and soc Ž.eR are simple for every primitive e s e g R. Hence it suffices by Lemma 4.6 to prove that R is artinian. If R has the ACC on right annihilators then R is left artinian by Proposition 4.5. Since this implies the ACC on left annihilators, R is also right artinian by symmetry. Now assume that RrsocŽ.R is right Goldie. Then R is semiprimary by Ž. Ž. Lemma 4.4, whence soc 2 R R s soc 2 RR . But R is right and left finite dimensionalŽ. by Theorem 3.14 so R is artinian bywx 1, Theorem 2.2 , as required.

COROLLARY 4.8. A right artinian ring R is quasi-Frobenius if and only if R is mininjecti¨e. Proof. If R is mininjective it is a minannihilator ring by Corollary 2.6, and so is a min-PF ring byŽ. 1 of Corollary 3.15. Hence R is quasi-Frobenius by Proposition 4.7.

PROPOSITION 4.9. Suppose that RrA is a min-PF ring for e¨ery ideal A of R. Then R is an artinian principal ideal ring.

Proof. By Theorem 3.14, socŽ.RrA is finitely generated and essential in RrA as a left and right module for each ideal A. Hence RrA is artinian bywx 3, Proposition 5 , and so is quasi-Frobenius. Thus, R is an artinian principal ideal ring bywx 10, p. 238 . 572 NICHOLSON AND YOUSIF

The next result is required below and has independent interest.

2 LEMMA 4.10. Suppose R is semiprimary and J s 0. Then R has ACC and DCC on left and right annihilators. Proof. Every ascending chain of left annihilators in R has the form Ž. Ž. lX12:lX :иии , where each Xiis a right ideal and X12= X = иии . 2 As J s 0, we have

J : lXŽ.Ž.12lJ :lXlJ :иии and X12q J = X q J = иии = J so, since RrJ is left noetherian and right artinian, there exists n G 1 such that иии иии lXŽ.Ž.nnlJ slXq1lJ s and Xnnq J s X q1q J s .

Now let k G n. Since Xkn: X and X nk: X q J, the modular law gives Ž. Ž. XnnsXlX kqJsX kqX nlJ. Thus,

lXŽ.nknslX Ž.llX ŽlJ .slX Ž. kkllX ŽlJ .

slXkkqŽ.Ž.XlJ slX k, proving the lemma.

2 THEOREM 4.11. If R is a semiprimary mininjecti¨e ring with JŽ. R s 0, then R is quasi-Frobenius. Proof. By Corollary 3.10, R is a minannihilator ring, so R is a min-PF ring. Hence R is quasi-Frobenius by Lemma 4.10 and Proposition 4.7.

If F is a field, the ring R FFis a right and left artinian ring with s 0 F 2 JRŽ.s0, but R is neither right nor left mininjective. Moreover, Camillo’s exampleŽ. see Remark 1.7 is a commutative semiprimary, local, mininjec- 3 tive ring with JRŽ.s0 which is not quasi-Frobenius.

Min-CS Rings A module M is called a Ž.min CS-module if every Ž simple . submodule of Mis essential in a summand of M. Haradawx 14 callsŽ. min CS-modules Ž.simple extending modules. A ring R is called a right Ž.min CS-ring if RR is aŽ. min CS-module. A module M is called Ž.min continuous if M is a Ž.min CS-module and everyŽ. minimal submodule of M that is isomorphic to a direct summand of M is itself a summand of M. For a full account of CS- and continuous modules, see Mohamed and Muller¨ wx 18 . Continuity was first introduced by Utumiwx 24 , who studied its relation to injectivity. He showed that an artinian, continuous ring is quasi-Frobenius. This result has been extended to rings with restricted chain conditions; see, for example, Ara and Parkwx 1 . MININJECTIVE RINGS 573

If F is a field, the ring R FFis left and right min-CS but is s 0 F neither left nor right mininjective because Srl­ S and S lr­ S . Our first result characterizes the right mininjective rings among the left min-CS rings.

PROPOSITION 4.12. Let R be a left min-CS ring. The following are equi¨alent:

Ž.1 R is right mininjecti¨e. Ž.2 kR simple, k g R, implies Rk simpleŽ R is right minsymmetric..

In particular, a commutati¨e min-CS ring is mininjecti¨e.

Proof. AlwaysŽ. 1 implies Ž. 2 . Conversely, if kR is simple, we must show ess 2 that lrŽ. k s Rk. Since Rk is simple by Ž. 2 , let Rk : Re, e s e,by ess min-CS. Then Rk : lrŽ. k : lr Ž Re .s Re,so Rk : lrŽ. k .As Rk is sim- Ž. Ž. ple, it suffices to show that lr k is semisimple, that is, that lr k : Sl. But if 0 / a g lrŽ. k then rk Ž.:ra Ž./R,sork Ž.sra Ž.because rk Ž.is Ž. maximal. Thus aR ( kR is simple, so Ra is simple by 2 and a g Sl,as required.

In what follows we investigate the class of semiperfect right continuous rings with essential right socle as an interesting generalization of the right PF-rings. The following lemma improves on Proposition 4.12.

LEMMA 4.13. Let R be a semiperfect left min-CS ring.

Ž. 1 If Slr: S then Rk simple, k g R, implies kR simple. Ž. 2 If Slrs S the following hold: Ž.i R is a left minannihilator ring. Ž.ii R is right mininjecti¨e. Ž.iii Rk is a simple left ideal if and only if kR is a simple right ideal. Ž. ess 3 If Srls S : R R then R is a right min-PF ring.

ess 2 Proof. Ž.1IfRk is simple let Rk : Re, where e s e g R. Then Re Ž. is indecomposable and Rk s soc Re s Sel . Since R is semiperfect, e is local and J q Ž.1 y eR is the unique maximal right ideal containing Ž.1yeR. It follows that rkŽ.Ž.:Jq1yeR, whence lr Ž.Ž. k = lJ l Ž. Ž. Re s Serl. But then Rk s Se:Ser:lr k by hypothesis, so rk= Ž . Ž. Ž. Ž. Ž . Ž. Ž . rSerr=rlr k s rk. Thus rksrSe so rk=Jq1yeR. It fol- lows that rkŽ.sJq Ž1yeR . is maximal, provingŽ. 1 . 574 NICHOLSON AND YOUSIF

Ž.2 Let Rk be a simple left ideal of R.If0/aglrŽ. k then rkŽ.:ra Ž.so aR is simple ŽŽ.rk is maximal byŽ.. 1 . It follows that Ž. Ž. ess Ž. lr k : Srl: S , whence lr k is a semisimple left ideal. But Rk : lr k by the proof of Proposition 4.12, so Rk s lrŽ. k , proving Ž. i . Now Ž. ii Ž. Ž . Ž. follows from Proposition 2.4 because Srl: S . Finally, ii implies iii by 1 and Theorem 1.14. Ž.3 This follows fromŽ. 2 by the definition of min-PF rings.

COROLLARY 4.14. Suppose that R is a semiperfect, min-CS ring with ess Srls S : R R. Then R is a min-PF ring. The con¨erse is true if R is a basic ring. Proof. As R is left min-CS, it is a right min-PF ring by Lemma 4.13, so 2 socŽ.Re is simple for all local e s e by Theorem 3.14. As R is left ess 2 min-CS, socŽ.Re : Rf for some f s f g R. Hence socŽ.Re s soc Ž.Rf . ess It follows that Re ( Rf by Theorem 3.16, and so socŽ.Re : Re. Hence ess SlR: R so, again, by Lemma 4.13, R is a left min-PF ring. Conversely, let R be a basic min-PF ring; we need only show that R is min-CS. Write 1 s e1 q иии qeni, where the e are basic orthogonal local idempotents. For each i, socŽ.Rei is simple Ž by Corollary 3.10. and essen- tial in Reii. But R is basic so the socŽ.Re are the only minimal left ideals in R by Theorem 3.6. Hence R is left min-CS; similarly R is right min-CS.

LEMMA 4.15. Let R be a semiperfect, left Kasch, left min-CS ring. Then: Ž. ess Ž. 1SlR: R and soc Re is simple and essential in Re for all local 2 esegR. Ž. 2 R is right Kasch if and only if Slr: S .

Ž.3IfÄ4 e1,...,en are basic local idempotents in R then Ä Ž. Ž.4 soc Re1 ,...,soc Ren is a complete set of distinct representati¨es of the simple left R-modules.

Proof. Given Ä4e1,...,en as inŽ. 3 , extend it to a set Ä4e1,...,enm,...,e of orthogonal local idempotents with 1 s e1 q иии qem.As Ris left Kasch, for each i s 1, 2, . . . , n, let ReiirJe ( K i: R, where K iis a simple left ess 2 ideal. Since R is a left min-CS ring, each K ii: Rf for some f iis f , and Ž. fi is local because R is semiperfect. We have soc Rfiiiis K ( Re rJe for Ž. Ž. is1, 2, . . . , n. Clearly Rfij( Rf if and only if soc Rfi( soc Rf j, if and Ä4 Ž. only if i s j. Thus Rf1,...,Rfniare pair-wise nonisomorphic with soc Rf simple and essential in Rfi for each i. Moreover,Ä socŽ.Rf1 ,...,soc Ž.Rfn 4is a complete set of representatives of the simple left R-modules. This proves Ž. 2 Ž. 3.If e segRis local then Re ( Rfi for some i so soc Re is simple and essential in Re. Furthermore S m socŽ.Re ess m Re R, liis [is1: [is1s provingŽ. 1 . MININJECTIVE RINGS 575

Ž. Ž. To prove 2 : If Slr: S then R is right Kasch by 1 andwx 19, Lemma 3 . Conversely, if R is right Kasch let K be a simple left ideal of R; we must ess 2 show that K : Sr . We have K : Re for some e s e g R, so it suffices Ž. to show that Ser /0. But e is local Re is indecomposable so Ser s d lJeŽ.( ŽeRreJ . is nonzero because R is right Kasch. LEMMA 4.16. Let R be a semiperfect, left continuous ring. Then: Ž. Ž . Ž . 1 ZRRR:JsZR. Ž. 2 Slr:S. Ž. ess 3 R is left Kasch if and only if SlR: R. In this case Ž.i R is also right Kasch. 2 Ž.ii soc ŽeR ./ 0 for all local e s e g R. 2 Ž.iii soc ŽRe . is simple and essential in Re for all local e s e g R. Ž. Ž . Ž . Proof. 1 ZRRR:Jin any semiperfect ring and J s ZRbywx 24 . Ž. Ž. 2 This follows from J s ZRR . Ž. ess 3IfRis left Kasch, SlR: R by Lemma 4.15. Conversely, if ess SlR:Rthen R is left Kasch bywx 1, Proposition 1.4 , and R is right Kasch byŽ. 2 and Lemma 4.15. Finally, as R is left Kasch we have 0 / Ž.Ž.d Ž.Ž. RerJe ( er J s eSlr: eS s soc eR by 2 . THEOREM 4.17. Let R be a semiperfect, left continuous ring with ess Ä4 SlR:R.If e1,...,en is a basic set of local idempotents in R, there exist elements t1,...,tn of R and a permutation ␴ of Ä41, 2, . . . , n such that:

Ž.1Ä4Ä4Rt1,...,Rtn and t1 R,...,tn R are complete sets of distinct repre- sentati¨es of the simple left and right R modules, respecti¨ely. Ž. Ž . Ž . 2 tRiii:soc e R and Rt s soc Re␴i for all i s 1, 2, . . . , n. Ž. 3 tRi (eR␴ i re␴ iiii J and Rt ( Re rJe for all i s 1, 2, . . . , n.

Proof. By Lemma 4.16, R is left Kasch and Slr: S . Hence, for any is1, 2, . . . , n. d 0 / Ž.Ž.ReiirJe ( er i J seS ilir:eS ssocŽ.eR i.

Hence choose a simple right ideal Tiil: eS. We have Tei␴i/0 for some Ä4 ␴ig1, 2, . . . , n , so let 0 / tiig Te␴iii. Thus tRsT is simple and tig Ž. Ž . eRei ␴ ii. Moreover, Rt is also simple because lti=JqR1ye i, a maxi- mal left ideal of R. But Rti : Re␴ i so, since R is a left min-CS ring, Ž. Rti s soc Re␴ ii. Now the maps re ¬ rtiand er␴ii¬tr are well defined epimorphisms Reiiª Rt and eR␴iiªtR, respectively, so ReiiirJe ( Rt s Ž. soc Re␴ i and eR␴ i reJ␴ ii(tR. Since the eiare basic, these results imply that both Ä4Ä4Rt1,...,Rtn and tR1 ,...,tRn are pairwise nonisomorphic, Ž. Ž. proving 1 . Moreover, soc Re␴ iii( Re rJe shows that ␴ is a permutation, completing the proof. 576 NICHOLSON AND YOUSIF

THEOREM 4.18. Let R be a semiperfect, left continuous ring with ess Ž. SlR:R.If R is also a right min-CS ring, then R is a right and left min-PF ring. ess Proof. If kR is a simple right ideal of R, then kR : eR for some local idempotent e of R by hypothesis. Since R is left Kasch, 0 / Ž.d Ž. RerJe ( er J s eSll: eR. Hence kR : eS : Sl,soSr:Sl. Thus Srs Sl by Lemma 4.16, and so R is a min-PF ring by Corollary 4.14.

5. UNIVERSALLY MININJECTIVE RINGS

It is well known that every right R-module is principally injective if and only if R isŽ. von Neumann regular. The analogous result for mininjectivity is as follows:

LEMMA 5.1. The following conditions are equi¨alent for a ring R: Ž.1 E¨ery right R-module is mininjecti¨e. Ž.2 E¨ery principal right R-module is mininjecti¨e. 2 Ž.3 K /0for e¨ery simple right ideal K of R. Ž. Ž . 4 soc RR l J s 0. Ž. Ž. 5 R is right mininjecti¨e and soc RR is projecti¨e as a right R- module. Proof. Ž.1 « Ž.2 and Ž. 3 m Ž.4 These are clear. Ž.2« Ž.3IfKskR is simple, k g R, we have an R-isomorphism ␥:kR ª RrrkŽ.given by ␥ Žka .s a q rk Ž..By2, Ž.␥ is left multiplication by c q rkŽ.for some c g R. Thus ck q rk Ž.s␥ Ž.ks1qrk Ž., whence 2 kck s k. Thus 0 / e s e g K, where e s kc. Ž. Ž. 3«1If␥:KªMR is R-linear where K is a simple right ideal, 2 thenŽ. 3 gives K s eR, e s e,so␥smиwhere m s ␥ Ž.e . Ž.3m Ž.5 Given Ž. 3 , Ž. 5 follows because each simple right ideal K 2 has the form K s eR, e s e. Conversely, if K is a simple right ideal, then 2 Ž. K(eR, where e s e because soc RR is projective. Since R is right mininjective, it follows that eR s cK for some c g R. Hence K ­ J and Ž.3 follows.

Call a ring R right uni¨ersally mininjecti¨e if it satisfies the conditions in Lemma 5.1. Clearly each ring with zero right socleŽ hence every polyno- mial ring. is right universally mininjective, and every semiprime ring is both right and left universally mininjective. On the other hand, a right universally mininjective ring R with essential right socle is semiprime by Ž.4 of Lemma 5.1. MININJECTIVE RINGS 577

Remark 5.1. While every right universally mininjective ring has projec- tive right socle, the converse is false: If F is a field, the ringFF has 0 F both socles projective but is neither right nor left mininjective. Remark 5.2. A direct product of rings is right universally mininjective if and only if each factor is right universally mininjective. If R is right universally mininjective and right Kasch, then R is semisim- ple artinian because every simple right module is projective. If R is I-finite we can say more.

THEOREM 5.2. IfRisI-finite then R is right uni¨ersally mininjecti¨e if and Ž. only if R ( S = Z, where S is semisimple artinian and soc ZZ s 0. Proof. If R is right universally mininjective then Theorem 1.12 gives Ž. Ž. R(S=Z, where S is semisimple artinian and soc ZZ : JZ. But Z is Ž. right universally mininjective by Remark 5.2, so soc ZZ s 0 by Lemma 5.1.

Theorem 5.2 is not new; it was proved by Gordonwx 12, Proposition 4.1 with a different proof. With only minor variations, the proof of Theorem 1.6 goes through to prove

THEOREM 5.3. Being right uni¨ersally mininjecti¨e is a Morita in¨ariant.

REFERENCES

1. P. Ara and J. K. Park, On continuous semiprimary rings, Comm. Algebra 19 Ž.1991 , 1945᎐1957. 2. Y. Baba and K. Oshiro, On a theorem of Fuller, J. Algebra 154 Ž.1993 , 86᎐94. 3. J. A. Beachy, On quasi-artinian rings, J. London Math. Soc. 3 Ž.1971 , 449᎐452. 4. J.-E. Bjork,¨ Rings satisfying certain chain conditions, J. Reine Angew. Math. 245 Ž.1970 , 63᎐73. 5. V. Camillo, Distributive modules, J. Algebra 36 Ž.1975 , 16᎐25. 6. V. Camillo, Commutative rings whose principal ideals are annihilators, Portugal. Math. 46 Ž.1989 , 33᎐37. 7. V. Camillo and M. F. Yousif, Continuous rings with acc on annihilators, Canad. Math. Bull. 34 Ž.1991 , 462᎐464. 8. C. W. Curtis and I. Reiner, ‘‘Representation Theory of Finite Groups and Associative Algebras,’’ Wiley᎐Interscience, New YorkrLondon, 1962. 9. J. Dieudonne,´ Remarks on quasi-Frobenius rings, Illinois J. Math. 2 Ž.1958 , 345᎐354. 10. C. Faith, ‘‘Algebra II, Ring Theory,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1976. 11. K. R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 Ž.1969 , 115᎐135. 578 NICHOLSON AND YOUSIF

12. R. Gordon, Rings in which minimal left ideals are projective, Pacific J. Math. 31 Ž.1969 , 679᎐692. 13. C. R. Hajarnavis and N. C. Norton, On dual rings and their modules, J. Algebra 93 Ž.1985 , 253᎐266. 14. M. Harada, On modules with extending properties, Osaka J. Math. 19 Ž.1982 , 203᎐215. 15. M. Harada, Self mini-injective rings, Osaka J. Math. 19 Ž.1982 , 587᎐597. 16. M. Harada, A characterization of QF-algebras, Osaka J. Math. 20 Ž.1983 , 1᎐4. 17. F. Kasch, ‘‘Modules and Rings,’’ London Mathematical Society Monographs, Vol. 17, Academic Press, New York, 1982. 18. S. H. Mohamed and B. J. Muller,¨ ‘‘Continuous and Discrete Modules,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 19. W. K. Nicholson and M. F. Yousif, Continuous rings with chain conditions, J. Pure Appl. Algebra 97 Ž.1994 , 325᎐332. 20. W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 Ž.1995 , 77᎐93. 21. B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 Ž.1966 , 373᎐389. 22. E. A. Rutter, Rings with the principal extension property, Comm. Algebra 3 Ž.1975 , 203᎐212. 23. H. H. Storrer, A note on quasi-Frobenius rings and ring epimorphisms, Canad. Math. Bull. 12 Ž.1969 , 287᎐291. 24. Y. Utumi, On continuous rings and self-injective rings. Trans. Amer. Math. Soc. 118 Ž.1965 , 158᎐173.