Journal of the
Korean Mathematical Society JKMS

Let $R$ be a ring with identity $1_{R}$ and let $U(R)$ denote the group of all units of $R$. A ring $R$ is called $locally$ $finite$ if every finite subset in it generates a finite semigroup multiplicatively. In this paper, some results are obtained as follows: (1) for any semilocal (hence semiperfect) ring $R$, $U(R)$ is a finite (resp. locally finite) group if and only if $R$ is a finite (resp. locally finite) ring; $U(R)$ is a locally finite group if and only if $U(M_{n}(R))$ is a locally finite group where $M_{n}(R)$ is the full matrix ring of $n \times n$ matrices over $R$ for any positive integer $n$; in addition, if $2 = 1_{R} + 1_{R}$ is a unit in $R$, then $U(R)$ is an abelian group if and only if $R$ is a commutative ring; (2) for any semiperfect ring $R$, if $E(R)$, the set of all idempotents in $R$, is commuting, then $R/J \cong \oplus_{i=1}^{m} D_{i}$ where each $D_{i}$ is a division ring for some positive integer $m$ and $|E(R)| = 2^{m}$; in addition, if $2 = 1_{R} + 1_{R}$ is a unit in $R$, then every idempotent is central.

Keywords: locally finite group, locally finite ring, semilocal ring, semiperfect ring, Burnside problem for matrix group, commuting idempotents

MSC numbers: Primary 16L30; Secondary 16U60