QF-3 endomorphism rings of Σ-quasi-projective modules

Abstract

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a 2-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and o [M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End( R M) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of o [M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].Throughout this paper R denotes an associative ring with identity, and /?-mod denotes the category of left R-modules. If M is a module, then we will say that a module N is M-generated (M-cogenerated) if it is a quotient (resp. a submodule) of a direct sum Af (/) (resp. direct product M 1 ) of copies of M. If N is M-cogenerated, then we will also say that N is M-torsionless and that M is a N-cogenerator. The full subcategory of /?-mod consisting of the submodules of M-generated modules will be denoted by o [M]; it is a locally finitely generated Grothendieck category [11]. We recall that a module N is M-projective (M-injective) if, for every quotient module (resp. submodule) X of M, the homomorphism Hom R (N, M)->Hom R (N, X) (resp. Hom R (M, N)-*Wom R {X, N)) is an epimorphism and, in particular, M is quasi-projective when it is M-projective. M is a projective object of a [M] precisely when it is Z-quasi-projective, that is, M (/) is quasi-projective for each set /. The largest M-generated submodule of a module X will be denoted by X M . E(N) will stand for an injective envelope of N in R-mod; if N belongs to o [M], then its injective envelope in this category is precisely E(N) M . A module is called finitely cogenerated (FC for short) if it has a finitely generated essential socle. When R R is injective and finitely cogenerated, R is said to be a left PF ring. The endomorphism ring of a module M will be denoted by S = End( R M) and we will use the convention of writing endomorphisms opposite scalars. We refer the reader to [2] and [6] for all the ring-theoretic notions used in the text.In [11], a module M is called a QF-3 module if there exists a minimal M-cogenerator t Work partially supported by the CAICYT (0784-84).Glasgow Math. J. 30 (1988) 215-220.https://www.cambridge.org/core/terms. https://doi