Korean J. Math. Vol. 21 No. 4 (2013) pp.463-471
DOI: https://doi.org/10.11568/kjm.2013.21.4.463

# 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).

## Supporting Agencies

Pusan National University

## References

 D. Dolz ̆an, Multiplicative sets of idempotents in a finite ring , J. Algebra 304 (2006), 271–277. Google Scholar

 H. K. Grover, D. Khurana and S. Singh, Rings with multiplicative sets of prim- itive idempotents, Comm. Algebra 37 (2009), 2583–2590. Google Scholar

 J. Han, Regular action in a ring with a finite number of orbits, Comm. Algebra 25 (1997), 2227–2236. Google Scholar

 J. Han, Group actions in a unit-regular ring, Comm. Algebra 27 (1999), 3353– 3361. Google Scholar

 J. Han and S. Park, An additive set of idempotents in rings , Comm. Algebra 40 (2012), 3551–3557. Google Scholar

 W. K. Nicholson, Introduction to Abstract Algebra, PWS Publishing Co., Boston, 1993. Google Scholar