A note on dual automorphism invariant modules

Abstract

In this paper, we investigate some properties of dual automorphism invariant modules over right perfect rings. Also, we introduce the notion of dual automorphism invariant cover and prove the existence of dual automorphism invariant cover. Moreover, we give the necessary and sufficient condition for every cyclic module to be a dual automorphism invariant module over a semi perfect ring and we prove that supplemented quasi projective module has finite exchange property. Also we give a characterization of a perfect ring using dual automorphism invariant module.

A Note on Automorphism Liftable Modules

“…In [5], Golan proved that a ring is perfect if and only if every module has a quasi projective cover. In [11], Selvaraj and Santhakumar characterized perfect ring by using dual automorphism invariant cover with certain condition. Also in [12], they proved that a ring is perfect if and only if every flat module is automorphism liftable.…”

“…Singh and Srivastava [13], introduced a new class of modules namely dual automorphism invariant modules, which is the dual notion of automorphism invariant modules introduced by Lee and Zhou [9]. Further study of such modules was carried out by various authors in various articles [1,8,11].…”

Automorphism liftable modules

“…The dual notion of automorphism-invariant modules was introduced by Singh and Srivastava in [11] and they called such modules as dual automorphism invariant modules. Further study on dual automorphism invariant modules was carried out in [8] and [10]. An R-submodule N of an R-module M is said to be small in M if N + K = M for any proper submodule K of M and it is denoted by N ≪ M .…”