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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. Q 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 Q 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 QF}3if ERŽ.R is projective. According to W. K. Nicholsonwx 17 , a ring R is called semiregu- lar 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, ;k 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 sum- mand 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 gen- erated 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 e¨ery primiti¨e 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 primiti¨e idempotents of R, there exist elements k1,...,kinRn ,and a permutation s of Ä41,...,n such that the following hold for all i s 1,...,n: Ž. a Rkii: Re and k i R : eRsŽi.. Ž. b Rki( ResŽi.rJesŽi. 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 eRsŽi. skRi seSsŽi. (eRiireJ. Ž.e Soc ŽRei . is homogeneous with each simple submodule isomor- phic to Res Ži.rJes 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.
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