Multiplicative set of idempotents in a semiperfect ring
Bull. Korean Math. Soc. 2011 Vol. 48, No. 5, 1033-1039
https://doi.org/10.4134/BKMS.2011.48.5.1033
Printed September 1, 2011
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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