Characterizing local rings via perfect and coperfect modules

TL;DR

This paper extends the concepts of perfect and coperfect modules using semidualizing modules over Noetherian rings, and characterizes local rings based on the existence of such modules.

Contribution

It introduces generalized notions of perfect and coperfect modules via classes associated with semidualizing modules and characterizes local rings through these modules.

Findings

Characterization of local rings using perfect modules.

Characterization of local rings using coperfect modules.

Development of properties and relations of these generalized modules.

Abstract

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the classes $ \mathcal{P}_C $ and $ \mathcal{I}_C $, we extend the notions of perfect and coperfect modules introduced by D.Rees \cite{R} and O.Jenda \cite{J1}. First, we study the basic properties of these modules and relations between them. Next, we characterize local rings in terms of the existence of special perfect (resp. coperfect) modules.