On Continuous Rings

JOURNAL OF ALGEBRA 191, 495᎐509Ž. 1997 ARTICLE NO. JA966936

On Continuous Rings

Mohamed F. Yousif*

Department of Mathematics, The Ohio State Uni¨ersity, Lima, Ohio 45804 and Centre de Recerca Matematica, Institut d’Estudis Catalans, Apartat 50, E-08193 Bellaterra, Spain View metadata, citation and similar papers at core.ac.uk brought to you by CORE

Communicated by Kent R. Fuller provided by Elsevier - Publisher Connector

Received March 11, 1996

We show that if R is a semiperfect ring with essential left socle and rlŽ. K s K for every small right ideal K of R, then R is right continuous. Accordingly some well-known classes of rings, such as dual rings and rings all of whose cyclic right R-modules are essentially embedded in projectives, are shown to be continuous. We also prove that a ring R has a perfect duality if and only if the dual of every simple right R-module is simple and R [ R is a left and right CS-module. In Sect. 2 of the paper we provide a characterization for semiperfect right self-injective rings in terms of the CS-condition. ᮊ 1997 Academic Press

According to S. K. Jain and S. Lopez-Permouth´ wx 15 , a ring R is called a right CEP-ring if every cyclic right R-module is essentially embedded in a projectiveŽ. free right R-module. In a recent and interesting article by J. L. Gomez´ Pardo and P. A. Guil Asensiowx 9 , right CEP-rings were shown to be right artinian. In this paper we will show that such rings are right

continuous, and so R is quasi-Frobenius if and only if MR2Ž.is a right CEP-ring. This result extends some of the work inwx 14 and wx 15 . We will also show that right CEP-rings inherit some of the important features which are known to hold for pseudo- and quasi-Frobenius rings, such asŽ. i Ž. Ž . Ž . Ž . Ž . Soc RRR s Soc R ,ii JRsZR RRsZR , iii R is left and right Kasch, andŽ. iv R admits a Nakayama permutation of its basic primitive idempotents. A ring R is called a D-ringŽ.Ž.Ž. dual ring if rl I s I and lr L s L for every right ideal I and every left ideal L of R. We will show that D-rings

*E-mail: [email protected].

495

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 496 MOHAMED F. YOUSIF are left and right continuous and as a result, R will have a perfect duality if and only if MR2Ž.is a D-ring. These results are direct consequences of a more general result which will be proved in Theorem 1.7 of this paper. We will show that if R is a ess Ž. semiperfect ring with Soc RRR : R and rl K s K for every small right ideal K of R then R is right continuous and rlŽ. A s A for every right ideal A of R. Motivated by the work of J. L. Gomez´ Pardo and P. A. Guil Asensiow 9, 10x we will also prove the following results:

Ž.A Ris a right CS-ring which is left and right Kasch if and only if R is semiperfect right continuous with essential right socle. Ž.B Ris a left and right CS-ring and the dual of every simple right R-module is simple if and only if R is semiperfect left and right continuous Ž. Ž. with Soc RRR s Soc R is essential as a left and as a right R-module in R. Moreover in this case R admits a Nakayama permutation of its basic set of primitive idempotents. Ž.C Rhas a perfect duality if and only if the dual of every simple right S-module is simple and S is a left and right CS-ring, where S s MR2Ž..

Following ideas of M. Haradawx 11 , K. Oshiro wx 20 , and D. v. Huynh wx13 we will show, in Sect. 2 of the paper, that R is semiperfect right self- injective if and only if R has DCC on principal projective right ideals, Ž. Ž . R[Ris a right CS-module, and JRsZRR . Throughout this paper all rings considered are associative with unity and all modules are unitary R-modules. We write A : BAŽ.;Bto mean A is a submoduleŽ. proper of B.If MR is a right R-module, we will denote by JMŽ., ZM Ž., Soc Ž.M , and EM Ž.the Jacobson radical, the singular submodule, the socle, and the injective hull of M, respectively. The left Ž.resp. right annihilator of a subset X of R is denoted by lX Ž.Žresp. rXŽ... We will write M Žk.to indicate to a direct sum of k-copies of M. ess The notation A : B and C :[ D will mean A is an essential submodule of B and C is a direct summand of D. We will indicate by R M* s HomŽ.MRR, R the dual left R-module. A module MR is said to satisfy:

the C1-conditionŽ. CS-condition if every submodule of M is essential in a summand of M; the C2-condition if every submodule of M which is isomorphic to a summand of M is itself a summand of M; CONTINUOUS RINGS 497

the C3-condition if M12and M are summands of M and M12l M s 0, then M12[ M is a summand of M.

MR is called continuous if M satisfies both the C1- and C2-conditions, and is called quasicontinuous if it satisfies the C1- and C3-conditions. A ring R is called a right CS-ringŽ. right continuous, right quasicontinuous if RR is a CS-moduleŽ. continuous module, quasicontinuous module . It is well known that M12[ M is continuous if and only if M1and M 2are continuous and Mijis M -injective, where i / j and 1 F i, j F 2. We refer the reader towx 4 andwx 16 for detailed information on continuous, quasicontinuous, and CS-modules. A ring R will be called left Kasch if every simple left

R-module is embedded in R, and is called right QFᎏ3if ERŽ.R is projective. According to W. K. Nicholsonwx 17 , a ring R is called semiregular if RrJRŽ.is a regular ring and idempotents lift modulo JRŽ..

1. CONTINUOUS RINGS

We begin with the following lemma which is of independent interest and has several interesting consequences which will be presentedŽ and dis- cussed. after we prove the main results of this section. Recall that a submodule N of a module M is said to lie over a direct summand of M if there exist submodules L and K of M such that M s L [ K, L : N, and NlK:JMŽ.. Ž. Ž . LEMMA 1.1. Suppose R is a semiregular ring with J R s ZRR .Then Žk. R satisfies the right C2-condition, ᭙k G 1. Proof. Since R is semiregular, every finitely generated submodule of Žk. R lies over a direct summand. Suppose ARR( B , where B is a summand of RŽk.. Then A is a finitely generated projective right R-module. Žk. Ž k. Let C and D be submodules of R such that R s C [ D, C : A, and Žk. AlDis a small submodule of R . Thus A s C [ Ž.A l D , where Ž Žk.. Ž.Žk. AlD:rad R s wZRR x. But this means A l D is a finitely generated projective singular right R-module. This is impossible, unless A l Žk. D s 0, and so A s C is a summand of R . ess ess LEMMA 1.2. Suppose R is a semiperfect ring, Soc RRR : R, Soc R RR: R , and rlŽ. K s K for e¨ery minimal right ideal K of R. Then the following hold:

Ž.i R is left mininjecti¨eiŽ.e., maps from minimal left ideals of R into R are gi¨en by right multiplication.. Ž.ii R is left and right Kasch. 498 MOHAMED F. YOUSIF

Ž.iii If k g R, Rk is a minimal left ideal of R if and only if kR is a minimal right ideal of R. Ž. Ž . Ž . iv Soc RRR s Soc R s S. Ž.v Soc ŽeR . is simple for eëry primitië idempotent e of R. Ž.vi R is right finite dimensional. Ž. Ž. Ž . Ž . vii JRsZRRRsZR . Ž. Ä4 viii if e1,...,en is a basic set of primitië idempotents of R, there exist elements k1,...,kinRn ,and a permutation ␴ of Ä41,...,n such that the following hold for all i s 1,...,n: Ž. a Rkii: Re and k i R : eR␴Ži.. Ž. b Rki( Re␴Ži.rJe␴Ži. and kiii R ( eRreJ.

Ž.c kR1 ,...,kn R and Rk1,...,Rkn are complete sets of representa- ti¨es of the simple right and simple left R-modules, respecti¨ely. Ž. Ž . d Soc eR␴Ži. skRi seS␴Ži. (eRiireJ.

Ž.e Soc ŽRei . is homogeneous with each simple submodule isomorphic to Re␴ Ži.rJe␴ Ži.. Proof. By Corollary 3.15 and Theorem 3.14 ofwx 19 . The next result was originally proved by Y. Utumiwx 24, Proposition 3.12 for left pseudo-Frobenius rings. With the help of the above two lemmas and by adapting Utumi’s argument we extend his result to a much larger class of rings. ess ess LEMMA 1.3. Suppose R is a semiperfect ring, Soc RRR : R, Soc R RR: R , ess and rlŽ. K s K for e¨ery minimal right ideal K of R. If A : rlŽ. A for some right ideal A of R, then ARR is essential in a summand of R .

Proof. Let ᎐: R ª RrJ denote the canonical quotient map, and let Ä4 RseR1 [иии [ eRn , where e1,...,en is a complete set of primitive Ž. orthogonal idempotents of R. By Lemma 1.2 above, S s Soc R R s SocŽ.RR and Soc Ž.eR is simple and essential in eR for every local idempotent e of R. Assume that the uniform dimension of AR is k, and so SocŽ.AR is the sum of k minimal right ideals of R, where k n.If ess F Ž. Ž. Ž. lA:J, then S s rJ:rl A , and so A : RR , and there is nothing to prove. Thus we may assume that lAŽ.oJ. Write lAŽ.sRŽ.1ye for some idempotent e of R. In this case, RŽ.1 y e q J s lA Ž.qJand if we apply the right annihilator to both sides, we get SocŽ.eR SocwrlŽ. A x. ess s Ž. Ž.Ž. Since A : rl A , we deduce that Soc AR s Soc eR s eS. Since R is semiperfect, SocŽ.AR is the sum of k minimal right ideals and SocŽ.eR is CONTINUOUS RINGS 499 simple and essential in eRi ,1FiFn, we infer that e is the sum of k primitive idempotents of R. Let x g lAŽ., such that x s Ž.1 y e . Then Ž.Ž.1yeyxgJ, and e q x has an inverse. Let t be an element of R with Ž.eqxts1. Then Ž.Ž. 1 y exts1ye. If we define ␴ : R Ž.Ž1 y e ª R 1 yex.ŽŽ..Ž.by ␴ r 1 y e s r 1 y ex,᭙rgR, then ␴ is a well-defined R-epimorphism. If rŽ.1 y exs0 for some r g R, then 0 s r Ž.1 y exts rŽ.1ye, and so ␴ is one-to-one. Now by Lemmas 1.1 and 1.2 above, it 2 follows that RŽ.1 y exsRf for some f s f g R. Since e is the sum of k primitive orthogonal idempotents of R, it follows thatŽ. 1 y e and hence f is the sum of Ž.n y k orthogonal primitive idempotents of R. Since Ž. Ž. Ž . Ž. xglA, it follows that A : rl A : 1 y fRand hence Soc AR : SocŽŽ 1 fR . .. Since both SocŽAR .and Soc ŽŽ 1 fR . . have the same right y ess y uniform dimension, we infer that A : Ž.1 y fR. LEMMA 1.4. The following conditions on a ring R are equiälent: Ž.i rl Ž K .s K for eëry small right ideal K of R. Ž.ii rRbw llKŽ.xsrbŽ.qK for eëry small right ideal K of R and eëry b g R. Proof. Ž.ii ª Ž.i Obvious. Ž.iª Žii . Clearly, Žrb Ž .qK .:rRbw llKŽ.xw. Suppose x g rRbllKŽ.x and y g lbKŽ.. Then ybK s 0 and so yb g Rb l lK Ž., which implies ybx s 0 and y g lbxŽ.. Thus lbx Ž.=lbK Ž .and hence bx g rl Ž. bx : rl Ž bK . s bK, since bK is a small right ideal of R. Thus bx s bk, for some b g K, i.e., Žx y k .g rb Ž.and x g K q rbŽ., which completes the proof. LEMMA 1.5. Suppose R is semiperfect and rlŽ. K s K for eëry small right ideal K of R. Then rlŽ. T s T, for eëry right ideal T of R. Proof. Since R is semiperfect, T lies over a direct summand of R, i.e.,

R s eR12[eR, such that eR1:T,TleR2is a small right ideal, and 22 Ž. e11seand e 22s e are idempotents of R. Thus T s eR1[TleR 2, Ž. Ž . Ž . Ž. Ž.Ž and so lT sR1ye12llTleR. Now, rl T s rRw 1ye1llTl .Ž. Ž. eR21xseRqTleR 2sT, by Lemma 1.4. Thus rl T s T, for every right ideal T of R. Ž. Ž . LEMMA 1.6. Suppose R is semiperfect with J R s ZRR .If e¨ery small right ideal of R is essential in a summand of R, then R is right continuous.

[ Proof. Let T be a right ideal of R. Write T A B, where A RR s ess[ : [ and B is a small right ideal of R. By hypotheses, B : C, where C : RR. By Lemma 1.1, since R satisfies the right C2-condition, and hence the ess Ž.[ right C3-condition, it follows that T : A [ C, where A [ C : RR. 500 MOHAMED F. YOUSIF

ess Ž. Ž. THEOREM 1.7. Suppose R is semiperfect, Soc RRR : R, and rl K s K for e¨ery small right ideal K of R. Then rlŽ. A s A for e¨ery right ideal A of R and R is right continuous. Proof. By Lemma 1.5, rlŽ. A A for every right ideal A of R. Byw 18, ess s Theorem 2.3x , Soc RRR: R . By Lemma 1.3, every right ideal A is essential in a summand. ByŽ. vii of Lemma 1.2 and Lemma 1.1, R satisfies the right C2-condition, and hence R is right continuous. According to Haradawx 12 , a ring R is called right simple-injective if every R-homomorphism from a right ideal of R into R with simple image is given by left multiplication by an element of R.

LEMMA 1.8. Suppose R is semiperfect right simple-injecti¨e with SocŽ.eR / 0 for e¨ery local idempotent e of R. Then R is right continuous. Proof. Bywx 19, Proposition 4.3 , R satisfies the hypotheses of Theorem 1.7. Hence R is right continuous.

Recall that a ring R is said to have perfect duality if RRR and R are injective cogenerators, or equivalently, if R is semiperfect left and right self-injective with essential rightŽ. left socle.

PROPOSITION 1.9.Ž. i E¨ery D-ring is left and right continuous. Ž.ii R has perfect duality if and only if M2Ž. R is a D-ring. Proof. PartŽ. i follows from Lemma 1.8 andwx 19, Theorem 4.1 , andŽ. ii follows fromŽ. i andwx 25, Corollary 7.5 .

PROPOSITION 1.10. Suppose R is a right CEP-ring. Then R is a right continuous ring and satisfies conclusions Ž.i ᎐ Žviii . of Lemma 1.2.

Proof. Clearly if I is a right ideal of R, then I s rlŽ. I . Byw 9, Corollary 3.3x , R is right artinian. Thus R is right continuous by Theorem 1.7. Obviously R satisfies the hypotheses and hence the conclusions of Lem- ma 1.2. The next corollary extends some of the work inwx 14 and wx 15 .

COROLLARY 1.11. A ring R is quasi-Frobenius if and only if M2Ž. R is a right CEP-ring. Proof. By Proposition 1.10 above andwx 25, Corollary 7.5 . Recall that a right R-module M is called weakly R n-injective if Ž. Ž. ᭙x1,..., xnRgEM there exists X : EM such that x1,..., xngX and n XRR(M.Mis called weakly injective if it is weakly R -injective for every nG1. If M is cyclic, then M is weakly injective if and only if it is weakly R2-injectiveŽ seewx 15. . CONTINUOUS RINGS 501

LEMMA 1.12. A ring R is right self-injectië if and only if R is right weakly Ž. Ž . injectië, semiregular and J R s ZRR . Ž. Ž. Proof. Suppose x g ERRRR. Then ᭚X : ER such that 1, x g X, Ž. and XRR( R . Since R satisfies the right C2-condition by Lemma 1.1 , XR also satisfies the right C2-condition. Thus RRRis a summand of X . ess Inasmuch as RRR: X , we infer that x g R and R is right self-injective. The converse is well knownwx 25 . PROPOSITION 1.13. Suppose R is a right CEP-ring. Then the following conditions are equiälent: Ž.i R is quasi-Frobenius. 2 Ž.ii R is right weakly R -injectië. Ž.iii Soc ŽRe . is simple for eëry local idempotent e of R. Ž.iv R is right mininjectiëi Ž.e., eëry R-homomorphism from a minimal right ideal of R into R is giën by left multiplication.. Ž.v The dual of eëry simple right R-module is simple. Proof. The implicationŽ. i l Žii . follows from Proposition 1.10,Ž. vii of Lemma 1.2, and Lemma 1.12. It is well known that a right artinian ring R Ž. Ž. Ž. is quasi-Frobenius if and only if Soc RRR s Soc R and both Soc eR and SocŽ.Re are simple for every local idempotent e of R. Now, by Proposition 1.10 andŽ. v of Lemma 1.2, Ž. i is equivalent to Ž iii . . The implication Ž.vª Ž.iv is obvious. To complete the proof we only need to show that Ž. iv impliesŽ. iii . But this follows from Proposition 1.10, Lemma 1.2, andw 19, Proposition 3.3x . Remark. If we replace conditionŽ. iii in Proposition 1.13 above by requiring the ring to be left finite dimensional, then R will be left artinian and need not be quasi-Frobenius. Being left artinian will follow from Proposition 1.10,Ž. iv of Lemma 1.2, andwxwx 3, Lemma 6 . In 2 , Bjork¨ has an example of a two-sided artinian local right CEP-ring which is not quasi- Frobenius. If R is a commutative ring then R is mininjective if and only if rlŽ. K s K for every minimal ideal K of R. We provide in the next proposition a large class of commutative continuous rings which are not necessarily self-injective. PROPOSITION 1.14. Suppose R is a commutatië semiperfect mininjectië ring with essential socle. Then R is a continuous ring. In particular eëry commutatië subdirectly irreducible local ring is continuous. Proof. ByŽ. vii of Lemma 1.2 and Lemma 1.1, R satisfies the C2-condition. ByŽ. v of Lemma 1.2, Soc ŽRe .is simple and essential in Re for every local idempotent e of R. Write R s Re1 [ иии [ Ren for a basic set of 502 MOHAMED F. YOUSIF

primitive idempotents Ä4e1,...,en of R. Let A be an ideal of R. Clearly AAeiiиии Ae for a subset Ä4Ä4e ii,...,e of e1,...,en,1 k n, s1[ [ k ess 1k F F and Ae / 0, 1 j k. Thus A Re иии Re [ R . Thus every iijF F : 1[ [ ik: R ideal is essential in a summand of R.

Recently, Gomez´ Pardo and Guil Asensiowx 10 proved that if R is right Kasch right CS-ring then SocŽ.RRRis finitely generated and essential in R . This remarkable result has some interesting consequences below.

LEMMA 1.15. E¨ery right Kasch ring satisfies the left C2-condition. 2 Proof. Suppose R is a right Kasch ring and L ( Re, where e s e g R and L is a left ideal of R. Then L s Ra, a g R, and laŽ.sR Ž1yf .for 2 some f s f g R. HenceŽ. 1 y fas0, a s fa, and aR : fR. It suffices to show that f g aR. For, if f s ab for some b g R, then aba s fa s a,so 2 LsRa s Rba, where Ž.ba s ba. Assume to the contrary f f aR. Then aR ; fR, so let aR : T ; fR, where T is a maximal submodule of fR.By right Kasch, let ␴ : fRrT ¨ R be an embedding, and put c s ␴ Ž.f q T . Since fa s a g T, ca s ␴ Ž.Ž.fa q T s ␴ a q T s 0. Thus c g la Ž.s 2 RŽ.1yf,so0scf s ␴ Žf q T .Žs ␴ f q T .s c, a contradiction.

THEOREM 1.16. The following conditions on a ring R are equi¨alent: Ž.i R is a right CS-ring which is left and right Kasch. Ž.ii R is right Kasch and a right continuous ring. Ž.iii R is a semiperfect right continuous ring with essential right socle.

Proof. Ž.i ª Žii . By Lemma 1.15, R satisfies the right C2-condition and so R is right continuous. Ž.ii ª Žiii . Bywx 10, Corollary 2.7 , R has a finitely generated essential right socle. In particular R has no infinite sets of orthogonal idempotents. By the work of Utumi inwx 25 , R is a semiregular ring and hence semiperfect. Ž.iii ª Ž.i Bywx 19, Lemma 4.16 . THEOREM 1.17. The following conditions on a ring R are equi¨alent: Ž.i R is a left and right Kasch left and right CS-ring. Ž.ii R is a left and right CS-ring, and the dual of e¨ery simple right R-module is simple. Ž. Ž. iii R is semiperfect left and right continuous ring with Soc R R s SocŽ.RR essential as a left and as a right R-module in R.

Moreoër, if R satisfies any of the aboë equiälent conditions then R satisfies the left as well as the right conclusions Ž.i ᎐ Žviii . of Lemma 1.2. CONTINUOUS RINGS 503

Ž. Ž . Ž . Proof. i ª iii By Theorem 1.16 we only need to show that Soc R R Ž. sSoc RR . But this can easily be verified since R is semiperfect left and Ž. Ž . Ž . right continuous, and hence JRsZRRRsZR . Ž.iii ª Ž.i By Theorem 1.16 Ž.ii ª Žiii . Since the dual of every simple right R-module is simple, R is a right Kasch ring. By Lemma 1.15, R satisfies the left C2-condition. Thus R is left continuous and hence semiregular bywx 25 . Now since R is right Kasch and a right CS-ring, it follows that R has no infinite sets of orthogonal idempotents bywx 10, Corollary 2.7 . Whence R is semiperfect. Ž. Ž. Ž . Thus Soc RRR : Soc R .If eis a local idempotent of R, then eRreJ * Ž. Ž . (lJ иesSoc RR и e is a simple left R-submodule of Re. Since R is a ess Ž. left CS-ring, Soc RR и e : Re, for every local idempotent e of R. This Ž. Ž . implies Soc Re s Soc RR . e is simple and essential in Re, for every local Ž. Ž. idempotent e of R. Thus Soc RRR s Soc R is essential as a left as well as a right R-module in R. Bywx 19, Lemma 4.16 , R is left Kasch, and by Lemma 1.15, R is right continuous. Ž.iii ª Ž.ii We only need to show that the dual of every simple right R-module is simple. If e is a local idempotent of R, then Ž.Ž.eRreJ * ( lJ и Ž. Ž. Ž. esSoc RRRи e s Soc R и e s Soc Re , which is simple and essential in Re, since Re is a left CS-module.

Now assume that R satisfies one of the above equivalent conditions. In order to show that R satisfies the leftŽ. right conclusions of Lemma 1.2 above, we must show that rlŽ. T s T and lr Ž K .s K for every minimal right ideal T and every minimal left ideal K of R. Since R is left and right Ž. Ž . Ž . Ž . Ž. continuous, JRsZRRRsZR . This implies that rl Soc R s Soc R . Now let T be a minimal right ideal of R. Since R is a right CS-ring, ess ess T:eR for some local idempotent e of R. Thus T : rlŽ. T : eR. Also ess T:SocŽ.R implies T : rlŽ. T : rl ŽSoc R .s Soc Ž.R , and hence T s rlŽ. T . By symmetry lrŽ. K s K for every minimal left ideal K of R.

THEOREM 1.18. The following conditions on a ring R are equi¨alent:

Ž.i R has a perfect duality. Ž.ii R is left and right Kasch and Ž R [ R . is a left and right CS-module. Ž.iii The dual of eëry simple right R-module is simple and Ž R [ Risa . left and right CS-module. Ž.iv The dual of eëry simple right S-module is simple and S is a right Ž. and left CS-ring, where S s M2 R the 2 = 2 matrix ring oër R. 504 MOHAMED F. YOUSIF

Proof. The implicationŽ. i ª Žii . and parts Ž iii . and Ž iv . are obvious. Ž.ii ª Ž.i By Lemma 1.15, R is left and right continuous, so R is Ž. Ž . Ž . semiregular with JRsZRRRsZR by Utumi’s work inwx 25 . Now, by Lemma 1.1 Ž.R [ R is left and right continuous as an R-module. Byw 16, Proposition 2.10x R is left and right self-injective ring. By a well-known result of B. Osofksywx 21 , R has a perfect duality. Ž.iii ª Ž.iv Bywx 4, Corollary 12.8 , S is a leftŽ. right CS-ring if and only if Ž.R [ R is a left Ž. right CS-module. By Theorem 1.17 and Lemma 1.2 above R is a leftŽ. right Kasch and left Ž. right mininjective ring. By Morita invariance, S is leftŽ. right Kasch, and bywx 19, Proposition 1.4 , S is left Ž.right mininjective. Thus the dual of every simple right Ž. left S-module is simple bywx 19, Proposition 2.2 . Ž.iv ª Ž.i By Theorem 1.17 above, S is semiperfect left and right Ž. Ž. continuous ring with Soc SSS s Soc S essential as a left and as a right S-module. Bywxwx 19, Lemma 3.17 and 25, Corollary 7.5 , R has a perfect duality. The next two results are consequences of Lemma 1.15.

COROLLARY 1.19. Suppose R is a left Kasch ring and Ł R is an arbitrary direct product of at least two copies of R. Then the following conditions are equi¨alent: Ž.i ŁR is a right CS-module. Ž.ii Ł R is injecti¨e as a right R-module. Proof. Ž.i ª Žii . Since R [ R is a right CS-module, it is right continuous by Lemmas 1.1 and 1.15, and so R is right self-injective. Thus Ł R is injective as a right R-module. Ž.ii ª Ž.i Clear. Recall that a ring R is called a rightŽ. countably ⌺-CS-ring if every direct sum of arbitraryŽ. countably many copies of R is CS as a right R-module. Right ⌺-CS-rings were also called right Co-H-rings by K. Oshirowx 20 and were shown to be left and right artinian. By Lemma 1.15 it is not difficult to see that if R is a left Kasch right ⌺-CS-ring, then R is right self-injective and hence quasi-Frobenius. Also if R is a right Kasch right ⌺-CS-ring then ERŽ.R is a right projective R-module by the work of Oshirowx 20 . Since ERŽ.R cogenerates all the simple right R-modules, R is a right PF-ring and hence quasi-Frobenius. For right countably ⌺-CS-rings we have the following result:

COROLLARY 1.20. A ring R is quasi-Frobenius if and only if R is right and left Kasch and a right countably ⌺-CS ring. Ž. Proof. Since R is left Kasch and R [ R R is a right CS-module, it follows from Lemmas 1.1 and 1.15 that R is a right self-injective ring. Thus CONTINUOUS RINGS 505

Ris a semiperfect ring by the work of Osofskywx 21 . By w 4, Corollary 8.11 x , R is right countably ⌺-injective. Thus R is quasi-Frobenius by a well-known result of C. Faithwx 6 . The converse is clear. Next we turn our attention to some of the applications of Lemma 1.1 as we promised earlier in the paper. Recall that a ring R is called right principally injective Ž.p-injective if every R-homomorphism from a principal right ideal of R into R is given by left multiplication by an element of R.

PROPOSITION 1.21.Ž. i A ring R is right continuous if and only if R is a Ž. Ž . semiregular right CS-ring with J R s ZRR . Ž.ii The following conditions on a ring R are equiälent: Ž.a R is right self-injectië. Ž. Ž . b R is right p-injectië and R [ RR is a right CS-module. Ž. Ž. c R is right continuous and R [ RR is a right CS-module. Ž.d R is semiregular, Ž.R [ R is a right CS-module and JŽ. R s ZRŽ.R . Proof. Ž.i Follows from Lemma 1.1. Ž.aª Ž.b Clear. Ž.bª Ž.c It is not difficult to see that if R is right p-injective then R satisfies the right C2-conditionŽ seewx 18, Theorem 1.2. . Thus R is right continuous. Ž.cª Ž.d Obvious. Ž. Ž. Ž. dªa By Lemma 1.1, R [ R R is a right continuous module, and, bywx 16, Proposition 2.10 , R is right self-injective. Remark. StatementŽ. i in the Proposition 1.21 extends the work of Mohamed and Muller¨ on right quasicontinuous ringsw 16, Proposition 3.15x .Ž. d of statement Ž. ii extends the work of Dung and Smith on regular ringswx 3, Proposition 3 and the work of K. Oshiro on right Co-H-rings Ž.right ⌺-CS-rings inwx 20, Theorem 4.3 . The next result is also of independent interest and generalizes some of the work by J. Rada and M. Saorin on semiregular right FGF-rings. A ring R is right FGF-ring if every finitely generated right R-module embeds in a free module. Our work depends on the recent discovery by J. L. Gomez´ Pardo and P. A. Guil Asensio that every right Kasch right CS-ring has finitely generated essential right soclewx 10 , and the following fact which Ž␣. can easily be verifiedwx 11 : RR is a CS-module if and only if every ␣-generated right R-module is a direct sum of a projective module and a singular module, where ␣ is any cardinal number. In fact our argument 506 MOHAMED F. YOUSIF below also shows the connection between some of the work of Rada and Saorinwx 22 and that of Gomez´ Pardo and Guil Asensio wx 9 . Ž. Ž . PROPOSITION 1.22. Suppose R is semiregular with J R s ZRR . Ž.i If eëry cyclic right R-module embeds in a free module, then R is right artinian and right continuous. Ž.ii If eëry 2-generated right R-module embeds in a free module, then R is quasi-Frobenius. Proof. Ž.iIfMis a cyclic right R-module, then M ( N, where N is a Žk. right R-submodule of R , for some k G 1. As we have seen before, we Žk. can write N s A [ B, where A, B : RR , A is a projective summand of RŽk., and B is a small submodule of RŽk.. Thus every cyclic right R-module is a direct sum of a projective module and a singular module, from which it follows that R is a right CS-ring. Thus R is right artinianwx 10 . By Theorem 1.7, R is right continuous.

Ž.ii As in the proof of Ž. i above, if MR is a 2-generated right R-module, then M is a direct sum of a projective module and a singular module. Thus RŽ2. is a right CS-module. By Proposition 1.21Ž. ii , R is right self-injective. Since R is also right Kasch, a well-known result of Osofsky wx21 asserts that R has finitely generated essential right socle. Thus every cyclic right R-module has finitely generated essential socle, from which we infer that R is right artinian and hence quasi-Frobenius.

2. SEMIPERFECT SELF-INJECTIVE RINGS

In this section we characterize semiperfect right self-injective rings in Ž. terms of R [ R R being a right CS-module. Our work here is motivated by that of Haradawx 11 , Oshiro wx 20 , and Huynh wx 13 on right ⌺-CS rings. THEOREM 2.1. The following conditions on a right R are equiälent: Ž.i R is semiperfect right self-injectië. Ž. Ž . ii R has DCC on principal projectië right ideals, R [ RR is a right Ž. Ž . CS-module, and J R s ZRR . Ž.iii R has DCC on principal projectië right ideals, is right quasicontin- Ž. uous, and R [ RR is a right CS-module. Proof. Ž.ii ª Ž.i Since R has DCC on projective principal right ideals and R is a right CS-ring, we infer that R has no infinite sets of orthogonal idempotents and so R is right finite dimensional. Again, since R is a right

CS-ring, we can write R s eR1 [иии [ eRni, where each eRis uniform. Ž. Let Eiis EeR,1FiFn. We claim that if a g Ei, then eR iqaR is projective and embeds in some eRt ,1FtFn. CONTINUOUS RINGS 507

Ž. Let ␴ : R [ R eRi qaR be an R-epimorphism. Since R [ R is a ª ess right CS-module, kerŽ.␴ : U, where U is a summand of Ž.R [ R . Write Ž. Ž . R[RsU[W, for some WRRi: R [ R . Then eRqaR ( Urker ␴ Ž. [W. Since Eiiis uniform and eRis projective, it follows that eRiqaR Ž. (Wand so eRi qaR is a uniform projective right R-module and is embedded in R R k eR, where each e Ä4e ,...,e . Suppose k [ s [is1ttiig 1n Ž. is the least integer such that there is an embedding ␴ of eRi qaR in k eR. Let T ␴ Ž.eR aR .If eR T/0 and eR T/0 for [is1tittils q l ml some l / m,1 l,m k, then Ž.Ž.eR T eR T/0, since T is a F F ttlml l l uniform right R-module. This is a contradiction. Thus eR T 0 for ti l s some i,1 i k. Assume without loss of generality that eR T 0 F F tk l s and let ␲ : k eR ky1eR be the natural projection. Then kt[is1ijª[js1 t Ž.ker ␲ T 0 and T is embedded in ky1eR, a contradiction. Thus ktl s [js1j Ž. ks1 and eRitqaR is embedded in some eR,1FtFn. We claim that for every i,1 i n,E eRfor some t Ä41,...,n. Suppose ᭚a E F F it( i ig 1g i Ž. such that a1 f eRi . Then there is an embedding ␴1 of eRi qaR1 in eR,1 t n. Consider the following diagram: t1 F 1 F

0 6

6 ␴1 6

0 eR aR e R i q 1 t1 inc.6

᭚␪ 1 6 Ei

By the injectivity of E , there exists an embedding ␪ of eRin E , such i 1 ti1 that eR eR aR ␪Ž.eR E.If␪ Ž.eR E, we are done. If ii; q 11: ti11: 1tis not, then we can repeat the above process and find an element a2 g Ei such that a ␪ Ž.eR. As before ᭚ an embedding ␴ of ␪ Ž.eR aR 21f t1 21t1q 2 in some eR,1 t n, and by the injectivity of E , ᭚ an embedding ␪ t2 F 2 F i 2 of eRin E such that eR eR aR ␪Ž.eR ␪ Ž.eR aR ti2 ii; q 11: t11; 1tq 2: ␪Ž.Ž.eR EeR. If this process continues, we will obtain an infinite 2 ti2: strictly ascending chain eR ␪Ž.eR ␪ Ž.eR иии EeR Ž.of sub- i ; 1 t12; 2 ti; : Ž. modules of EeRiteach of which is isomorphic to some eR,1FtFn. Thus ᭚i and j such that i / j and ␪ Ž.eR ␪ Ž.eR and ␪ Ž.eR itij; jt it i( ␪Ž.eR eRfor some t Ä41,...,n. Thus eRproperly contains a copy jtj ( t g t of itself, which produces an infinite strictly descending chain of principal projective right ideals of R in eRti, a contradiction. Thus EeRŽ.is finitely generated and hence isomorphic to some eR,1 t n. In particular tiiF F ERŽ. nEeRŽ.is finitely generated and projective and so R is a Ris[is1 508 MOHAMED F. YOUSIF

␴i right QF-3-ring. Since for each i, E EeRŽ.eR,1 i n,isa iits ( i FF uniform injective module, it follows that End RiŽ.E is a local ring. By the projectivity of Eii, we infer that each E is a local moduleŽ i.e., has a unique .Ž. maximal submodule . Finally, we claim that EeRiiseR, for all i g Ä4 Ž. Ž. 1,...,n. Suppose EeRii/eR for some i. Then eRi:JE iand so ␴Ž.eR eJR Ž.Ž.ZR , a contradiction with the projectivity of eR. ii: ti : R i Thus eRi is injective, 1 F i F n, and R is a right self-injective ring. By right self-injectivity, R is a semiregular ring. Since R has no infinite sets of orthogonal idempotents, R is a semiperfect ring. Ž.i ª Žii . and Ž iii . Suppose R is a semiperfect right self-injective ring. It is enough to show that R has DCC on principal projective right ideals.

Suppose aR12>aR>иии is an infinite strictly descending chain of principal projective right ideals aRi of R. Since R is right self-injective, each aRiiseRfor some idempotent eiof R. Thus eR12>eR>иии , a contradiction of the fact that R has no infinite sets of orthogonal idempotents. Ž . Ž. Ž . Ž. iii ª i As in the proof of ii ª i above, R s eR1 [иии [ eRn , each eRis right uniform and EeRŽ.eRfor some t Ä41,...,n.If ii( tiig ti, then eRis injective. If t / i, then eRis eR-injective, since R is iis iiti right quasicontinuouswx 16, Corollary 2.14 . Thus eRi is a summand of Ž. Ž. EeRiiiiand hence eRsEeR and again eRis injective. Thus R is right self-injective. Inasmuch as R is also right finite dimensional, we infer that R is a semiperfect ring.

Remarks.iIfŽ. RsZthe ring of integers, then R is a commutative Ž. Ž. Ž . noetherian quasicontinuous ring with JRsZRs0 and R [ R R is a CS-module. However, R is not injective or semiperfect. Ž.ii Inwx 2 , Bjork¨ has an example of a local left and right artinian right Ž. Ž . Ž . Ž . continuous ring with JRsZRRRsZR . However, R [ R is not left or right CS as an R-module. Ž. FF iii If R s wx0 F , where F is a field, then R is left and right artinian, 2 JRŽ.s0, and R [ R is CS as a left and right R-module. However, R is not a left or right self-injective ring.

ACKNOWLEDGMENTS

This research was carried out during the academic year 1995᎐1996 while the author was spending a sabbatical leave at Centre de Recerca MatematicaŽ. CRM , Institut d’Estudis Catalans, Barcelona, Spain. The author gratefully acknowledges the financial support of the CRM and the Spanish Ministerio de Educacion y Ciencia. The author would also like to thank Professors Pere Ara, Keith Nicholson, and Dolors Herbera for many helpful conversa- tions, suggestions, and comments. CONTINUOUS RINGS 509

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