Modules over the endomorphism ring of a finitely generated projective module

Abstract

Let PR be a projective module with trace ideal T. An R-module XR is T-accessible if XT = X. If PR is finitely generated projective and C is the R-endomorphism ring of PR, such that CPR, then for XR, Hom (PR, XR)C is artinian (noetherian) if and only if XR satisfies the minimum (maximum) condition on T-accessible submodules. Further, if XR is T-accessible then Hom (PR, XR)C is finitely generated if and only if XR is finitely generated. The purpose of the present paper is to investigate cHom (CPR, XR), where PR is a finitely generated projective R-module and C = End (PR), the R-endomorphism ring of P, with respect to the properties of chain conditions and finite generation. Throughout this paper R is a ring with identity and all modules over R are unitary. The convention of writing module-homomorphisms on the side opposite the scalars is adopted here. 1. Preliminaries. Let PR be a finitely generated projective R-module with C = End (PR) such that CPR. The dual module of PR is (with respect to RR), RPC = Hom (CPR, RRR). It is well known, see [1], that the map PR 6P PR* = Hom (RP*, RR) given by p p, where ffp = fp, forf E P* is an R-isomorphism. LEMMA 1.1. For PR finitely generated projective, C = End (PR), the map End (PR) -End (RP*) given by c -c j, where fc= fc is a ring isomorphism. PROOF. The above map is nothing more than the composite of the following maps Hom (1, 6p) Hom (PR, PR) RHom (PR pR*)