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Generalizations of Principally Injective Ringsu JOURNAL OF ALGEBRA 206, 706᎐721Ž. 1998 ARTICLE NO. JA977403 Generalizations of Principally Injective RingsU Stanley S. Page The Uni¨ersity of British Columbia, Vancou¨er, British Columbia, V6T 1Z2, Canada E-mail: [email protected] and Yiqiang Zhou Memorial Uni¨ersity of Newfoundland, St. John's, Newfoundland, A1C 5S7, Canada E-mail: [email protected] Communicated by Kent R. Fuller Received December 20, 1997 A ring R is said to be right P-injective if every homomorphism of a principal right ideal to R is given by left multiplication by an element of R. This is equivalent to saying that lrŽ.a s Ra for every a g R, where l and r are the left and right annihilators, respectively. We generalize this to only requiring that for each 0 / a g R, lrŽ.a contains Ra as a direct summand. Such rings are called right AP-injective rings. Even more generally, if for each 0 / a g R there exists an n ) 0 with ann/ 0 such that Ra is not small in lrŽan., R will be called a right QGP-injective ring. Among the results for right QGP-injective rings we are able to show that the radical is contained in the right singular ideal and is the singular ideal with a mild additional assumption. We show that the right socle is contained in the left socle for semiperfect right QGP-injective rings. We give a decomposition of a right QGP-injective ring, with one additional assumption, into a semisimple ring and a ring with square zero right socle. In the third section we explore, among other things, matrix rings which are AP-injective, giving necessary and sufficient conditions for a matrix ring to be an AP-injective ring. ᮊ 1998 Academic Press INTRODUCTION Throughout we assume R is a ring with identity and modules are unitary right R-modules. Let JRŽ.be the Jacobson radical of R and ZM ŽR .be * The research was supported by NSERC Grants A80204 and OGP0194196. 706 0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. PRINCIPALLY INJECTIVE RINGS 707 the singular submodule of an R-module M. The Baer criterion for testing when an R-module M is injective naturally leads one to consider the case when one can extend any homomorphism from any principal right ideal of R to M to a homomorphism of R to M. A module M is said to be principally injective, or P-injective for short, if every map from any principal right ideal to M extends to a map of R to M, i.e., is given by left multiplication by an element of M. This is equivalent to saying that lrMRŽ.a s Ma for all a g R, where l and r are the left and right annihila- tors, respectively. When the ring is P-injective as a right module, the ring is said to be a right P-injective ring. Many results which are known for right self-injective rings, such as the equality of the Jacobson radical with the right singular ideal, and in some cases equality of the right and left socle of R, seewx 2, 6 , were shown to hold for right P-injective rings in w 7, 9x . The second author began to study the rings for which one could write each homomorphism from a principal right ideal to R as the sum of two homomorphisms in a non-trivial way, with the first term a map which did extend to all of R and the second term one which could not be extended. More precisely, suppose the module M has the property that for all 0 / a g R, lrMRŽ.a s Ma [ Xa, where this is a direct sum over the endomorphism of ring of M. Such modules are called AŽ. almost P-injec- tive modules. Many of the results ofwx 9 were obtainable for the class of right AP-injective rings. This suggested the possibility one could obtain these same results if for all 0 / a g R, Ra is not small in lrŽ.a as a left R-module. Indeed, this turns out to be the case and is the subject of Section 2 where we consider the QŽ. quasi P-injective rings and even a more general setting, QGP-injective rings. In Section 3 we extend the results on maximal right ideals of Camillowx 3 on commutative rings andw 9, 10x for right P-injective rings to AP-injective rings. Also, we study the categorical aspects of AP-injective rings and give necessary and sufficient conditions for a matrix ring over a right AP-injective ring to be right AP-injective. Section 1 gives the notation and relation with annihilators which are needed in Sections 2 and 3 as well as examples to illustrate the concepts. 1. NOTIONS AND EXAMPLES For a right R-module M, we let S s EndŽ.M and then we have an Ž.S, R -bimodule M. For X : M and A : R, the right annihilator of X in R is rRŽ.X s Ä4r g R : Xr s 0 and the left annihilator of A in M is l M Ž.A s Ä4x g M : xA s 0 . For singletons Äx4Äand a4, we abbreviate to rRMŽ.x and l Ž.a . We may omit the subscripts R and M if there is no chance for ambiguity. For an R-module N and a submodule P of N,we 708 PAGE AND ZHOU will identify HomŽ.N, M with the set of maps in Hom Ž.P, M that can be extended to N, and hence HomŽ.N, M becomes a left S-submodule of HomŽ.P, M . In particular, for an element a g R,Hom Ž.R, M will be regarded as a left S-submodule of HomŽ.aR, M . DEFINITION 1.1. Given a module MR we let S s EndŽ.M . The module M is said to be almost principally injecti¨e Ž.or AP-injecti¨e for short if, for any a g R, there exists an S-submodule XaMof M such that lrŽŽ..Ra s Ma [ Xa as left S-modules. The module M is called almost general principally injecti¨e Ž.or AGP-injecti¨e for short if, for any 0 / a g R, there exists a positive integer n s naŽ.and an S-submodule Xa of M such that n n n a / 0 and lrMRŽŽa .. s Ma [ Xaas left S-modules. In each case the module Xaamay not be unique, but we take one such X for each a and form the S-module bMŽ.s ÝaaX . We call bMŽ.an index bound of M and the set of those XaRan index set of M.If R is an AP-injectiveŽ resp. AGP-injective.Ž module, then we call R a right AP-injecti¨e resp. right AGP-injecti¨e. ring. The concepts of AP-injective modules and AGP-injective modules are explained by the following lemma. LEMMA 1.2. Let MR be a module, S s EndŽ.M , and a g R. Ž.1 If lrMR Ž Ž..a s Ma [ X for some X : M as left S-modules, then we ha¨e Hom RRŽ.aR, M s Hom Ž.R, M [ ⌫ as left S-modules, where ⌫ s Ä f g Hom RŽ.Ž.aR, M : fag X 4; Ž.2 If Hom RR ŽaR, M .s Hom ŽR, M .[ ⌫ as left S-modules, then lrMRŽŽ..a s Ma [ X as left S-modules, where X s Ä faŽ.: f g ⌫4; Ž.3 Ma is a summand of lrMRŽ Ž..a as left S-modules iff Hom RŽ.R, M is a summand of Hom RŽ.aR, M as left S-modules. Proof. The map ␪ : lrMRŽŽ..a ª Hom R ŽaR, M .with ␪ Ž.m s ␭mis an S-isomorphism, where ␭mm: aR ª M is defined by ␭ Ž.ar s mr. Also, ␪Ž.Ma s Hom ŽR, M .. So, the lemma follows. Moreover, SŽ.Ma is non- small in lrMRŽŽ..a if and only if HomŽ.R, M is non-small in HomŽ.aR, M . From Lemma 1.2, the following lemma follows. LEMMA 1.3. For a g R and a module MRMR, lrŽŽ..a s Ma if and only if e¨ery R-homomorphism of aR into M extends to R. Recall that a module MR is called P-injectiveŽ. principally injective if every R-homomorphism from a principal right ideal to M is given by left multiplication by an element of M, while a module NR is said to be GP-injectiveŽ. generally principally injective if, for any 0 / a g R, there PRINCIPALLY INJECTIVE RINGS 709 exists a positive integer n such that an / 0 and any R-homomorphism of anR into N extends to R. Therefore, by Lemma 1.3, we have LEMMA 1.4.Ž. 1 A module MR is P-injecti¨e iff M is AP-injecti¨ewithan Ž.or each index bound b Ž.Ž. M s 0; Ž.2 A module MR is GP-injecti¨e iff M is AGP-injecti¨e with an Ž or each.Ž index bound b M .Žs 0.. We fix the following notation. If N is a submodule of M, we write NM< to indicate that N is a direct summand of M. For an Ž.R, R -bimodule M, we let R A M be the trivial extension of R and M, i.e., R A M s R [ M as an abelian group, with the following multiplication: Ž.Ž.Žr, xs, y s rs, ry q xs .
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