A finite additive set of idempotents in rings

Abstract

Let $R$ be a ring with identity $1$, $I(R) \neq \{0\}$ 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 $e, f \in I(R)$ $(e \neq f)$, $e + f \in I(R)$. In this paper, the following are shown: (1) $I(R)$ is a finite additive set if and only if $M(R) \setminus \{0\}$ 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).