Portal de Periódicos da CAPES

On the endomorphism ring of a free module

1983; Autonomous University of Barcelona; Volume: 27; Linguagem: Inglês

10.5565/publmat_27183_03

2014-4350

P. Menal,

Commutative Algebra and Its Applications

Throughout, let R be an (associative) ring (with 1) .Let F be the free right R-module, over an infinite set C, with endomorphism ring H .In this note we first study those rings R such that H is left coherent .By comparison with Lenzing's characterization of those rings R such that H is right coherent [8, Satz 41, we obtain a large class of rings H which are right but not left coherent .Also we are concerned with the rings R such that H is either right (left) IF-ring or else right (left) self-FP-injective .In particular we prove that H is right self-FP-injective if and only if R is quasi-Frobenius (QF) (this is an slight generalization of results of Faith and .Walker [31 which assure that R must be QF whenever H is right self-i .njective) moreover, this occurs if and only if H is a left IF-ring .On the other hand we shall see that if ' R is .pseudo-Frobenius(PF), that is R is an .injective cogeneratortin Mod-R, then H is left self-FP-injective .Hence any PF-ring, R, that is not QF is such that H is left but not right self FP-injective .A left R-module M is said to be FP-injective if every R-homomorphism N -.M, where N is a -finitély generated submodule of a free module F, may be extended to F .In other words M is FP-injective if and only if Ext l (K,M)=0 for every finitely presented module K .R is said to be 1eft self-FP-injective A right R-module is said to be torsionless if it is contai :ed in a direct product of copies of R .If M is a right R-module, we denote by F1 the torsionless module associated to M, that is ii=M/N, where N= n Kert t EHom(M,R)