t-Nilpotent invariant modules

ABSTRACT

We introduce the notion of a t-nilpotent endomorphism of modules and study the modules which are invariant under t-nilpotent endomorphisms of their injective envelope, such as modules are called t-nilpotent invariant. We provide several characterizations and investigate properties of each of these concepts. It is shown that a module M is t-nilpotent invariant if and only if $M=Z_2\left(M\right)\oplus M^{\prime }$, Z₂(M) is quasi-injective, M^′ is nilpotent invariant and Z₂(M) is M^′-injective. Also, it is proved that, for modules, being t-nilpotent invariant is a Morita invariant property. Moreover, we provide some characterizations of known rings in terms of t-nilpotent invariant modules.

KEYWORDS:

• Nilpotent-invariant module
• quasi-injective module
• t-automorphism invariant module
• t-nilpotent invariant module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D50
• 16D70