Multiplicative set of idempotents in a semiperfect ring
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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.
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.
참고문헌 (4)
- V. P. Camillo, "Exchange rings, units and idempotents", Comm. Algebra, 22(12), pp.4737~4749 (1994)
- D. Dolzan, "Multiplicative sets of idempotents in a finite ring", J. Algebra, 304(1), pp.271~277 (2006)
- J. Han, "Extensions of clean rings", Comm. Algebra, 29(6), pp.2589~2595 (2001)
- T. Y. Lam, "A First Course in Noncommutative Rings" (1991), Springer-Verlag