Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(4): 993-1006

Online first article July 9, 2019      Printed July 31, 2019

https://doi.org/10.4134/BKMS.b180759

Copyright © The Korean Mathematical Society.

On reversibility related to idempotents

Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung Ju Ryu, Hyo Jin Sung, Sang Jo Yun

Pusan National University; Pusan National University; Daejin University; Dong-A University; Pusan National University; Pusan National University; Dong-A University

Abstract

This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring $R$ shall be said to be {\it quasi-reversible} if $0\neq ab\in I(R)$ for $a, b\in R$ implies $ba\in I(R)$, where $I(R)$ is the set of all idempotents in $R$. We investigate the quasi-reversibility of $2$ by $2$ full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.

Keywords: quasi-reversible ring, Abelian ring, reversible ring, matrix ring, polynomial ring, NI ring

MSC numbers: 16U80, 16S36, 16S50

Supported by: The first named author was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korean Government (MSIP)(NRF-2017R1A5A1015722). The third named author was supported by the National Natural Science Foundation of China(11361063)