Modules which are invariant under nilpotents of their envelopes and covers

Abstract

© 2020 World Scientific Publishing Co. Pte Ltd. All rights reserved. A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Kosan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775-2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module M over a semiprimary ring is nilpotentinvariant iff all nilpotent endomorphisms of submodules of M extend to nilpotent endomorphisms of M. It is also shown that a right module M over a prime right Goldie ring with dim(M/Z(M)) > 1 is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if M is a finitely generated nilpotent-coinvariant right module with M/J(M) square-full, then M is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.