THE STRUCTURE OF SEMIPERFECT RINGS

Title & Authors
THE STRUCTURE OF SEMIPERFECT RINGS
Han, Jun-Cheol;

Abstract
Let R be a ring with identity $\small{1_R}$ 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 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$\small{(M_n(R))}$ is a locally finite group where $\small{M_n(R)}$ is the full matrix ring of $\small{n{\times}n}$ matrices over R for any positive integer n; in addition, if $\small{2=1_R+1_R}$ 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 $\small{R/J\cong\oplus_{i=1}^mD_i}$ where each $\small{D_i}$ is a division ring for some positive integer m and |E(R)|=$\small{2^m}$; in addition, if 2=$\small{1_R+1_R}$ 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;
Language
English
Cited by
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