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초록· 키워드
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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.
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참고문헌 (7)
참고문헌 신청
C. Huh / 2004 / Examples of strongly ¼-regular rings / Journal of Pure and Applied Algebra 189 (1) : 195 ~ 210
I. N. Herstein / 1976 / Noncommutative Rings, The Mathematical Association of America / John Wisley and Sons, Inc.
J. Cohen / 1988 / The group of units in a compact ring / Journal of Pure and Applied Algebra 54 (2) : 167 ~ 179
J. Cohen / 1992 / The structure of compact rings / Journal of Pure and Applied Algebra 77 (2) : 117 ~ 129
K. E. Eldridge / 1967 / D:C:C: rings with a cyclic group of units / Duke Mathematical Journal 34 : 243 ~ 248
I. N. Herstein / 1976 / Noncommutative Rings, The Mathematical Association of America / John Wisley and Sons, Inc.
J. Cohen / 1988 / The group of units in a compact ring / Journal of Pure and Applied Algebra 54 (2) : 167 ~ 179
J. Cohen / 1992 / The structure of compact rings / Journal of Pure and Applied Algebra 77 (2) : 117 ~ 129
K. E. Eldridge / 1967 / D:C:C: rings with a cyclic group of units / Duke Mathematical Journal 34 : 243 ~ 248
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