Nilpotent-invariant modules and rings

Automorphism-invariant modules, due to Lee and Zhou, generalize the notion of quasi-injective modules. A module which is invariant under automorphisms of its injective hull is called an automorphism-invariant module. Here we carry out a study of the module which is invariant under nilpotent endomorphisms of its injective envelope, such as modules are called nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that (1) nilpotent-invariant modules have the (C3) property, (2) if M = M-1 circle plus M-2 is nilpotent-invariant, then M-1 and M-2 are relative injective. In this paper, we also show that (3) a simple right nilpotent-invariant ring R is either right self-injective or R-R is uniform square-free.