Rings of quotients of endomorphism rings

Abstract

Introduction.The problem of describing the projective modules R M such that their endomorphism ring S = End(RM) satisfies some specific property has been considered in many papers. A frequent approach consists in defining a module-theoretic analogue of a ring-theoretic property and trying to prove that these modules provide the desired solution. But this approach is somewhat naYve, for in the passage from R to M too many things are lost: in categorical terms R is, among other things, a finitely generated generator of its module category R-mod but M is not. One of the first papers that deals with these questions is [i~]. Therein, Ware succesfully solves the problem for semiperfect and perfect endomorphism rings but the regular modules that he defines give only a sufficient condition (in the finitely generated case) for S to be a (Von Neumann) regular ring, which illustrates the drawbacks of this method. In [7] and [8] a rather general technique has been introduced to find necessary and sufficient conditions on M for S to have a specific property. In fact, since M is not supposed to be a generator, we do not need to work with all the category R-rood, which is replaced by the Grothendieck category o[M ] of all the modules subgenerated by M (i.e., submodules of quotients of direct sums of M). This makes unnecessary the projectivity of M and we will only assume that M is E-quasi-projective (i.e., that all direct sums of copies of M are quasi-projective or, equivalently, M is a projective object of o[M]). If M is, furthermore, finitely generated, then this condition reduces to quasi-projectivity. When M is a finitely generated quasi-projective selfgenerator (M generates all its submodules), then HomR(M,-) induces a category equivalence between ~[M] and S-mod by a result of Fuller [5]. This is a sort of Morita equivalence "relative to M" but the hypotheses on M are still very strong. In general, M is not a generator of a[M] (a self-generator) due to the fact that there are in d[M] modules in which M has zero trace, that is, the class TM= {Xec[M]IHomR(M,X) = O} can have nonzero modules. But we may get a generator in the following way: consider the quotient category ~[M]/T M (in the sense of Gabriel's localization theory [6]) of o[M] modulo the hereditary torsion class T M. Then o[M]/T M is a Grothendieck category, there is an exact canonical functor O:a[M] , c[M]/T M and Q(M) is a projective generator of o[M]/T M whose endomorphism ring as an object of this category is canoni-*Work partially supported by the CAICYT.