Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2011; 48(5): 1033-1039

Printed September 1, 2011

https://doi.org/10.4134/BKMS.2011.48.5.1033

Copyright © The Korean Mathematical Society.

Multiplicative set of idempotents in a semiperfect ring

Sangwon Park and Juncheol Han

Dong-A University, Pusan National University

Abstract

Let $R$ be a ring with identity $1$, $I(R)$ be the set of all idempotents in $R$ and $G$ be the group of all units of $R$. In this paper, we show that for any semiperfect ring $R$ in which $2 = 1 + 1$ is a unit, $I(R)$ is closed under multiplication if and only if $R$ is a direct sum of local rings if and only if the set of all minimal idempotents in $R$ is closed under multiplication and $eGe$ is contained in the group of units of $eRe$. In particular, for a left Artinian ring in which $2$ is a unit, $R$ is a direct sum of local rings if and only if the set of all minimal idempotents in $R$ is closed under multiplication.

Keywords: semiperfect ring, minimal idempotents

MSC numbers: Primary 16L30; Secondary 16U60

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