On a class of modules

Abstract

Let R be a ring with identity and M a unital right R-module. Let Z*(M) = {m ∈ M : mR ≪l E(mR)}. In this study we consider the property (T): For every right R-module M with Z* (M) = Rad M, M is injective. We give a characterization of the property (T) when R is a prime PI-ring. Also, over a right Noetherian ring R we prove that if R satisfies (T) then every right R-module is the direct sum of an injective module and a Max-module.