Topic
Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
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TL;DR: The t-class semigroup of an integral domain R, denoted St(R), is the semigroup which modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication.
13 citations
TL;DR: The non-commutative version of this theorem is unsettled as discussed by the authors, and it is not known whether a group with zero is a R-semigroup unless it admits a ring structure.
Abstract: A multiplicative semigroup S with 0 is said to be a R-semigroup if S admits a ring structure. Isbell proved that if a finitely generated commutative semigroup is a R-semigroup, then it should be finite. The non-commutative version of this theorem is unsettled. This paper considers semigroups, not necessarily commutative, which are principally generated as a right ideal by single elements and semigroups which are generated by two independent generators and describes their structure. We also prove that if a cancellative 0-simple semigroup containing an identity is a R-semigroup, then it should be a group with zero.
13 citations
TL;DR: In this article, it was shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute and the smallest integer m is the smallest one with the above property.
13 citations
13 citations
01 Jan 1983
TL;DR: In this paper, a characterization of perfect semigroup rings is given by means of the properties of the ring A and the semigroup G. This characterization works in arbitrary characteristic and is a natural strengthening of the conditions for A(G) to be semilocal.
Abstract: A characterization of perfect semigroup rings A (G) is given by means of the properties of the ring A and the semigroup G. It was proved in (10) that for a ring with unity A and a group G the group ring A(G) is perfect if and only if A is perfect and G is finite. Some results on perfectness of semigroup rings were obtained by Domanov (3). He reduced the problem of describing perfect semigroup rings A(G) to checking that certain semigroup algebras derived from A(G) satisfy polynomial identities. Further, a characterization of such PI-algebras over a field of characteristic zero was found in (2). However, the obtained results are difficult to formulate and refer to some exterior constructions obscuring an insight into the properties of the semigroup. The purpose of this paper is to completely characterize perfect semigroup rings by means of the properties of the semigroup and the coefficient ring. Our approach is quite different from that of (3) and omits PI-methods. It works in arbitrary characteristic and the final result is a natural strengthening of the conditions for A(G) to be semilocal (7). In what follows A will be an associative ring, G-a semigroup. A is said to be
13 citations