- MULTIPLICATIVE SET OF IDEMPOTENTS IN A SEMIPERFECT RING
- Park, Sang-Won ; Han, Jun-Cheol ;
- Bulletin of the Korean Mathematical Society, volume 48, issue 5, 2011, Pages 1033~1039
- DOI : 10.4134/BKMS.2011.48.5.1033

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.