Endomorphism rings of self-generators

Birge Huisgen-Zimmermann

Vol. 61 (1975), No. 2, 587–602

DOI: 10.2140/pjm.1975.61.587

Abstract

The group of R-homomorphisms Hom_R(M,A), where M, A are modules over a ring R, is, in a natural way, a module over the endomorphism ring S of M. Under certain weak assumptions on M, the following is true: Hom_R(M,−) carries injective envelopes of R-modules into injective envelopes of S-modules iff M generates all its submodules. Modules of the latter type are called self-generators. For M a self-generator, Hom_R(M,−) has additional properties concerning chain conditions and the socle. Many of the known results in this area, in particular those for M projective, are special cases of our main theorems.

Mathematical Subject Classification

Primary: 16A42, 16A42

Milestones

Received: 15 May 1975

Published: 1 December 1975