Abstract

A module which is invariant under automorphisms of its injective envelope is called an automorphism-invariant module. The class of automorphism-invariant modules was introduced and investigated by Lee and Zhou in 2013. In this paper, we study the class of modules which are invariant under all nilpotent endomorphisms of their injective envelopes of index two, such as modules are called 2-nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that a nonsingular module M is a weak duo 2-nilpotent-invariant module if and only if End(E(M)) is a strongly regular ring. For the ring R satisfying every cyclic right R-module is 2-nilpotent-invariant, we prove that R≅R1×R2, where R1,R2 are rings which satisfy R1 is a semi-simple Artinian ring and R2 is square-free as a right R2-module and all idempotents of R2 is central.