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Finitely Generated Multiplicative Subsemigroups of Rings
Journal article   Peer reviewed

Finitely Generated Multiplicative Subsemigroups of Rings

D.D Anderson
Semigroup forum, Vol.55(3), pp.294-298
11/1997
DOI: 10.1007/PL00005930

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Abstract

Isbell showed that if R is a commutative ring (not necessarily with identity) with multiplicative semigroup (R, .) finitely generated, then R is actually finite. Let Z(R) and U(R) denote the zero divisors and units of R, respectively. We show that the multiplicative subsemigroup (Z(R), .) of (R, .) (respectively, (R -U(R), .)) is finitely generated if and only if R is finite or R is an integral domain (respectively, a field).

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