Showing papers on "Cancellative semigroup published in 1986"
53 citations
TL;DR: In this paper, the authors describe the structure of WA in the singular case and the criterion for distributivity of the M -modules WCd' since the necessary conditions for that established in [ 1: 8.51 limit d tod
25 citations
TL;DR: In this article, the authors give a review of the results concerning the description of the Jacobson radical of semigroup rings of commutative semigroups and show that these results are also valid for other concrete radicals.
Abstract: The aim of this paper is to give a review of the results concerning the description of the Jacobson radical of semigroup rings of commutative semigroups. As an application we study when such semigroup rings are semilocal or local. In the final section we show that the results on the Jacobson radical are also valid for other concrete radicals.
8 citations
TL;DR: In this paper it was shown that a semigroup satisfies the permutation property of degree n (n ≥ 2) if every product of n elements inS remains invariant under some nontrivial permutation of its factors.
Abstract: A semigroupS satisfiesPPn, thepermutation property of degree n (n≥2) if every product ofn elements inS remains invariant under some nontrivial permutation of its factors. It is shown that a semigroup satisfiesPP3 if and only if it contains at most one nontrivial commutator. Further a regular semigroup is a semilattice ofPP3 right or left groups, and a subdirect product ofPP3 semigroups of a simple type. A negative answer to a question posed by Restivo and Reutenauer is provided by a suitablePP3 group.
8 citations
7 citations
TL;DR: Theorem 3.3 as mentioned in this paper states that a semigroup ring is Artian if and only if R is unitary and S is not idempotent, while Theorem 2.3 states that R[S] is noetherian if S and (R, + ), the additive group of R, are finitely generated.
6 citations
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TL;DR: In this paper, it was shown that on a regular semigroup 5, the relation E restricted to RC(5) is a congruence and a characterisation of the greatest element of each E-class has been presented.
Abstract: If S is an inverse semigroup then E is a congruence on C(S). If S is a regular semigroup then each E -class of C(5) is a complete modular sublattice of C(5). (See [6]. ) In [5, Sec. 3] Petrich presents a few characterisations of E when S is an inverse semigroup. Here we prove that on a regular semigroup 5, the relation E restricted to RC(5) is a congruence. Also we extend Petrichfs results to the lattice RC(S) of a regular semigroup 5 and present a characterisation of the greatest element of each E-class. Characterisations of the least element of each E-class have been presented by Feigenbaum [I] and La Torte[4]. THE LATTICE RC(S) We use, whenever possible, the notation of Howie [3]. Recall first that a regular semigroup 5 is said to be R-u_~potent if its set of idempotents E(S) is a left reqular band, i.e. if E(5) satisfies the identity ere = el. In [7,1; 8, 1.1 ] it is shown that on a regular semigroup 5,
TL;DR: In this article, necessary and sufficient conditions for the units of a commutative semigroup ring R[S] to be determined by the nilradical of R [S] and the unit of R[G] where G is the group of units of S is a torsion-free semigroup.
Abstract: We seek necessary and sufficient conditions for the units of a commutative semigroup ring R[S] to be determined by the nilradical of R[S] and the units of R[G] where G is the group of units of S. We assume that R is a commutative ring with identity and S is a torsion-free semigroup with identity.
TL;DR: In this article, it was shown that a semidirect product of an inverse semigroup and a group, in that order, contains an inverse subsemigroup that is a retract and that, together with the retraction mapping, forms free inverse morphic image of the semi-direct product.
Abstract: It is shown that a semidirect product of an inverse semigroup and a group, in that order, contains an inverse subsemigroup that is a retract and that, together with the retraction mapping, forms free inverse morphic image of the semidirect product. The congruence determined by the retraction mapping is shown to be determined by the semigroup of idempotents of the semidirect product.