THE STRUCTURE OF SEMIPERFECT RINGS

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Let R be a ring with identity <TEX>$1_R$</TEX> 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 semi group 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<TEX>$(M_n(R))$</TEX> is a locally finite group where <TEX>$M_n(R)$</TEX> is the full matrix ring of <TEX>$n{\times}n$</TEX> matrices over R for any positive integer n; in addition, if <TEX>$2=1_R+1_R$</TEX> 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 <TEX>$R/J\cong\oplus_{i=1}^mD_i$</TEX> where each <TEX>$D_i$</TEX> is a division ring for some positive integer m and |E(R)|=<TEX>$2^m$</TEX>; in addition, if 2=<TEX>$1_R+1_R$</TEX> is a unit in R, then every idempotent is central.