A Finite Additive Set of Idempotents in Rings
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Let R be a ring with identity 1, I(R) ≠ ｛0｝be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e+f ∈ I(R). In this paper, the following areshown: (1) I(R) is a finite additive set if and only if M(R)＼｛0｝ is acomplete set of primitive central idempotents, char(R) = ∈ and everynonzero idempotent of R can be expressed as a sum of orthogonalprimitive idempotents of R; (2) for a regular ring R such that I(R)is a finite additive set, if the multiplicative group of all units of Ris abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite fields).
Abstract
Let R be a ring with identity 1, I(R) ≠ ｛0｝be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e+f ∈ I(R). In this paper, the following areshown: (1) I(R) is a finite additive set if and only if M(R)＼｛0｝ is acomplete set of primitive central idempotents, char(R) = ∈ and everynonzero idempotent of R can be expressed as a sum of orthogonalprimitive idempotents of R; (2) for a regular ring R such that I(R)is a finite additive set, if the multiplicative group of all units of Ris abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite fields).
참고문헌 (6)
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