WEAK GENERATORS FOR CLASSES OF R-MODULES

Abstract

Let R be a ring. An R-module M is called a weak generator for a class C of R-modules if HomR(M,V) is non-zero for every non-zero module V in C. A projective module M is a weak generator for C if and only if M = MA for every annihilator A of a non-zero module V in C. Given any class C of R-modules, a finitely annihilated R-module M is a weak generator for the class of injective hulls of modules in C if and only if the R-module R/A is a weak generator forC, whereA is the annihilator of M. Moreover a finitely annihilated R-module M is a weak generator for the class of all injective R-modules if and only if the annihilator of M is a left T-nilpotent ideal. In case the ring R is commutative, a finitely generated R-module M is a weak generator for the class of all R-modules if and only if M is a weak generator for the class of injective R-modules. In addition, if the ring R is Morita equivalent to a commutative semiprime Noetherian ring, then M is a weak generator for the class of all R-modules if and only if the trace of M in R is an essential right ideal of R.