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 $2 Keywords locally finite group;locally finite ring;semilocal ring;semiperfect ring;Burnside problem for matrix group;commuting idempotents; Language English Cited by References 1. J. Cohen and K. Koh, The group of units in a compact ring, J. Pure Appl. Algebra 54 (1988), no. 2-3, 167-179 2. J. Cohen and K. Koh, The structure of compact rings, J. Pure Appl. Algebra 77 (1992), no. 2, 117-129 3. K. E. Eldridge and I. Fischer, D.C.C. rings with a cyclic group of units, Duke Math. J. 34 (1967), 243-248 4. I. N. Herstein, Noncommutative Rings, The Mathematical Association of America, John Wisley and Sons, Inc., 1976 5. C. Huh, N. K. Kim, and Y. Lee, Examples of strongly${\pi}\$-regular rings, J. Pure Appl. Algebra 189 (2004), no. 1-3, 195-210

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