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.
Sangwon Park and Juncheol Han
Dong-A University, Pusan National University
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
2020; 57(3): 691-707
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd