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Rings Whose Modules Are {Oplus}-Supplemented Journal of Algebra 218, 470᎐487Ž. 1999 Article ID jabr.1998.7830, available online at http:rrwww.idealibrary.com on Rings Whose Modules Are [-Supplemented Derya Keskin Department of Mathematics, Uni¨ersity of Hacettepe, 06532 Beytepe, Ankara, Turkey View metadata, citation and similar papers at core.ac.uk brought to you by CORE E-mail: [email protected] provided by Elsevier - Publisher Connector Patrick F. Smith Department of Mathematics, Uni¨ersity of Glasgow, Glasgow G12 8QW, Scotland E-mail: [email protected] and Weimin Xue* Department of Mathematics, Uni¨ersity of Iowa, Iowa City, Iowa 52242 Communicated by Kent R. Fuller Received November 19, 1998 We prove that a ring R is serial if and only if every finitely presented right and left R-module is [-supplemented, and that R is artinian serial if and only if every right and left R-module is [-supplemented. ᮊ 1999 Academic Press 1. INTRODUCTION AND PRELIMINARIES Throughout this paper we assume that R is an associative ring with identity and all R-modules are unitary right R-modules, unless otherwise specified. TheŽ. Jacobson radical of R is denoted by J. An R-module M is uniserial if its submodules are linearly ordered by inclusion and it is serial if it is a direct sum of uniserial submodules. The ring R is rightŽ. left serial if the right Ž. left R-module RRRR Ž.is serial and it is serial if it is both right and left serial. *Current address: Fujian Normal University, Fuzhou, Fujian 350007, People's Republic of China. 470 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. [-SUPPLEMENTED MODULES 471 Let M be an R-module. A submodule N of M is superfluousŽ. small if N q L / M for every proper submodule L of M. The notation N < M means that N is a superfluous submodule of M. M is called lifting Žor satisfies Ž..D1 if for every submodule N of M there are submodules KЈ and K of M such that M s K [ KЈ, KЈ F N, and N l K < K. It is easy to see that every uniserial module is lifting. Vanaja and Puravwx VP proved that a ring R has the property that all right R-modules are lifting if and only if R is an artinian serial ring with J 2 s 0. Also, Oshiro and Wisbauer obtained this result as a corollary in wxOW . As a generalization of lifting modules, Mohamed and MullerÈ wx MM call an R-module M [-supplemented if for every submodule N of M there is a summand K of M such that M s N q K and N l K < K. It was shown inwx KHS, Theorem 1.4 that a finite direct sum of [-supplemented modules is [-supplemented. In this paper we study rings whose modules are [-supplemented. In Section 2, we show that every f.g.Ž. finitely generated right R-module is [-supplemented if and only if every cyclic right R-module is [-supple- mented and every f.g. right R-module is a direct sum of cyclic modules. Arbitrary direct sums of lifting right R-modules over a right perfect ring R are shown to be [-supplemented. In Section 3, we prove that R is serial if and only if every f.p.Ž. finitely presented right R-module and f.p. left R-module is [-supplemented. Rings whose f.g. right and f.g. left modules are [-supplemented are also characterized. This class of rings properly contains noetherian serial rings. As stated in the Abstract, we show that R is artinian serial if and only if every right and left R-module is [-supple- mented. However, we note that an artinian left serial ring R need not be right serial although every right R-module is [-supplemented. For characterizations of [-supplemented modules and lifting modules we refer towx MM and wx Wi . Also, for the other definition and notation in this paper we refer towx AF . 2. [-SUPPLEMENTED MODULES THEOREM 2.1. The following statements are equi¨alent for a ring R with radical J. Ž.1 R is semiperfect. Ž.2 E¨ery f. g. free R-module is [-supplemented. [ Ž.3 RisR -supplemented. Ž.4 For e¨ery maximal right ideal A of R there exists an idempotent e g R y A such that A l eR : J. Ž.5 Any of the left-handed ¨ersions of Ž.Ž.2, 3 or Ž.4. 472 KESKIN, SMITH, AND XUE Proof. Ž.1 « Ž.2 Let R be a semiperfect ring. Let F be a f.g. free R-module. Byw MM, Theorem 4.41Ž. 2 and Proposition 4.8x , F is [-supple- mented. Ž.2 « Ž.3 Clear. Ž.3 « Ž.4 Let A be a maximal right ideal of R. There exists a direct summand K of R such that R s A q K and A l K < K. There exists an idempotent e in R such that K s eR. Clearly e f A. Moreover, A l K < l : RR so that A K J. Ž.4 « Ž.1 Let U be any simple R-module. Let 0 / u g U and B s Är g Rur< s 04 . Then B is a maximal right ideal of R and U ( RrB.By Ž.4 , there exists an idempotent f g R y B such that B l fR : J. Clearly R s B q fR. Moreover, B l fR < R implies that B l fR < fR. Now fRrŽ.Ž.B l fR ( B q fR rB s RrB ( U. It follows that U has a projec- tive cover. Bywx AF, Theorem 27.6 , R is semiperfect. Ž.1 m Ž.5 By symmetry. COROLLARY 2.2. A commutati¨e ring R is semiperfect if and only if e¨ery cyclic R-module is [-supplemented. Proof. Ž.¥ By Theorem 2.1. Ž.« Let I be any ideal of R. Then the factor ring R s RrI is still a semiperfect ring. By Theorem 2.1, R is [-supplemented as an R-module and hence [-supplemented as an R-module. Thus every cyclic R-module is [-supplemented. THEOREM 2.3. Let R be any ring and let M be a f. g. R-module such that e¨ery direct summand of M is [-supplemented. Then M is a direct sum of cyclic modules. s q иии q Proof. Suppose that M mR1 mRk for some positive integer k g F F s and elements mi M Ž.1 i k .If k 1 then there is nothing to prove. Suppose that k ) 1 and that the result holds for Ž.k y 1 -generated mod- ules with the stated condition. There exist submodules K, KЈ of M such s [ Ј s q l < Ј ( that M K K , M mR11K, and mR K K. Note that K r s q r ( r l Ј Ž.ŽM K mR11K .K mR ŽmR1K ., so that K is cyclic. On r l ( q r s r the other hand, K Ž.Ž.mR1111K mR K mR M mR, so that r l y l < K Ž.Ž.mR11K is k 1 -generated. Since mR K K it follows that K is Ž.k y 1 -generated. By induction, K is a direct sum of cyclic modules. Thus M s K [ KЈ is a direct sum of cyclic modules. Using the proof of Theorem 2.3, we have COROLLARY 2.4. Let R be a ring. Then e¨ery 2-generated [-supplemented R-module is a direct sum of cyclic modules. [-SUPPLEMENTED MODULES 473 COROLLARY 2.5. Let R be a ring and let n be a positi¨e integer. Then e¨ery n-generated R-module is [-supplemented if and only if Ž.i e¨ery cyclic R-module is [-supplemented, and Ž.ii e¨ery n-generated R-module is a direct sum of cyclic modules. Proof. Ž.« By Theorem 2.3, since every direct summand of an n-gen- erated module is n-generated. Ž.¥ Bywx KHS, Theorem 1.4 . COROLLARY 2.6. Let R be a ring. Then e¨ery f. g. R-module is [-sup- plemented if and only if Ž.i e¨ery cyclic R-module is [-supplemented, and Ž.ii e¨ery f. g. R-module is a direct sum of cyclic modules. A commutative ring R is called an FGC ring if every f.g. R-module is a direct sum of cyclic modules. FGC rings are discussed by Brandal where he gives a complete characterizationwx B, Theorem 9.1 . It is easy to give an example of a semiperfect ring which is not FGC. Let F be any field and R s FXww, Y xx, the ring of formal power series over F in the indeterminates X, Y. Then R is a commutative noetherian local domain and thus is semiperfect. However, the ideal J s RX q RY is the unique maximal ideal of R and is uniform, so is not a direct sum of cyclic modules. Thus J is not a [-supplemented R-moduleŽ. Corollary 2.4 and R is not an FGC ring. The following definitions are given inwx B , and we recall them for the convenience of the reader: A family of sets is said to have the finite intersection property if the intersection of every finite subfamily is non-empty. An R-module M is q linearly compact if whenever Ä4miiiM g I is a family of cosets of submod- g F g ules of MmŽ.iiM and M M for each i I with the finite intersec- F q tion property, then ig IiŽ.m M iis non-empty.
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