A FINITE ADDITIVE SET OF IDEMPOTENTS IN RINGS

Let R be a ring with identity 1, I(R) not equal {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 is an element of I(R) (e not equal f), e+f is an element of I(R). In this paper, the following are shown: (1) I(R) is a finite additive set if and only if M(R) \ {0} is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive 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 R is abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite fields).