The structure of semiperfect rings

Abstract

Let R be a ring with identity 1R 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(Mn(R)) is a locally finite group where Mn(R) is the full matrix ring of n × n matrices over R for any positive integer n; in addition, if 2 = 1R + 1R 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 [수식] where each Di is a division ring for some positive integer m and |E(R)| = 2m; in addition, if 2 = 1R + 1R is a unit in R, then every idempotent is central.