LEFT COTORSION RINGS

Abstract

It is proved that if R is an associative ring that is cotorsion as a left module over itself, and J is the Jacobson radical of R, then the quotient ring R/J is a left self-injective von Neumann regular ring and idempotents lift modulo J. In particular, if R is indecomposable, then it is a local ring. Let R be an associative ring with identity. A left R-module RM is called cotorsion if Ext 1(F, M) = 0 for every flat left R-module RF . Cotorsion modules were introduced by Harrison [4], and are a generalization of pure-injective modules. Their interest derives from the recent result of Bican, El Bachir and Enochs (see [1] and [8, Theorem 3.4.6]), that every module admits a cotorsion envelope. The ring R is called left cotorsion if it is cotorsion when considered as a left R-module over itself. Left cotorsion rings are closely related to pure-injectivity from another point of view. The category R -Flat of flat left R-modules is a locally finitely presented additive category, and therefore admits a theory of purity, as propounded in [2]. By [5, Lemma 3], R is pure-injective as an object of R -Flat if and only if it is left cotorsion. The point of this paper is to extend to these kinds of rings, results of Zimmermann and Zimmermann-Huisgen [10] about rings that are pure-injective as left modules over themselves. It is proved (see Theorem 6) that if R is a left cotorsion ring and J is the Jacobson radical of R, then the quotient ring R/J is a von Neumann regular ring. Furthermore, we show (see Corollary 9) that R/J is injective as a left module over itself, and that idempotents lift modulo J (see Corollary 4). Every left pure-injective ring R may be realized as the endomorphism ring of an injective object of some Grothendieck category, so these results about left cotorsion rings may be seen as generalizations of classical results concerning the endomorphism ring of an injective module. However, let us remark that our results are a proper extension, since there exist many left cotorsion rings that are not left pure-injective. Take, for example, any left perfect ring that is not left pureinjective (see [9]). We should also point out that, as the endomorphism ring of any injective object of a Grothendieck category is always left pure-injective, standard functor category techniques cannot be used in the study of left cotorsion rings. We develop in this paper new techniques based on the behaviour of direct limits, whose ¯ ¯