# CHARACTERIZING ABELIAN GENERALIZED REGULAR RINGS THAT ARE NOETHERIAN

• Han, Juncheol (Department of Mathematics Education, Pusan National University) ;
• Sim, Hyo-Seob (Department of Applied Mathematics, Pukyong National University)
• Received : 2019.10.28
• Accepted : 2020.01.09
• Published : 2020.01.31

#### Abstract

A ring R is called generalized regular if for every nonzero x in R there exists y in R such that xy is a nonzero idempotent. In this paper, we observe some equivalent conditions for the generalized regular rings that are abelian in terms of idempotents, and we also investigate the primitivity of an idempotent for such a ring. By using the investigation, we characterize such a kind of rings that are noetherian by showing that an abelian generalized regular ring R is noetherian if and only if R is isomorphic to a direct product of finitely many division rings. We also observe some interesting consequences of our results.

#### Acknowledgement

Supported by : Pukyong National University

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