On regular semigroup rings

1984 ◽  
Vol 99 (1-2) ◽  
pp. 145-151 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Regular Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Semigroup Rings ◽  
Principal Ideals ◽  
Algebra Over A Field

SynopsisIt is shown that if A is an algebra over a field, then the regularity of the semigroup algebra A[G] implies that the semigroup G is periodic. This enables us to characterize regular semigroup algebras of semigroups with d.c.c. on principal ideals. Also, regular self-injective semigroup algebras are described.

scholarly journals Self-injectivity of semigroup algebras

2020 ◽  
Vol 18 (1) ◽  
pp. 333-352
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Regular Semigroup ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Sufficient And Necessary Conditions ◽  
Finite Regular Semigroup

Abstract It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

Semiprimeness of semigroup algebras

2021 ◽  
Vol 19 (1) ◽  
pp. 803-832
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Regular Semigroup ◽  
Regularity Condition ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Locally Finite ◽  
Regular Semigroups ◽  
Abundant Semigroups ◽  
Sufficient And Necessary Conditions

Abstract Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

Locally ample semigroup algebras

2014 ◽  
Vol 07 (04) ◽  
pp. 1450067 ◽  
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Ample Semigroup

An idempotent-connected abundant semigroup S is a locally ample semigroup if for any idempotent e of S, the local submonoid eSe of S is an ample subsemigroup of S. Clearly, an ample semigroup is a locally ample semigroup. In this paper, it is proved that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semigroup algebra of an associate primitive abundant semigroup. As an application of this result, we characterize Jacobson radicals of finite locally ample semigroup algebras.

THE FIRST COHOMOLOGY GROUP OF BANACH INVERSE SEMIGROUP ALGEBRAS WITH COEFFICIENTS IN -EMBEDDED BANACH BIMODULES

2019 ◽  
Vol 101 (3) ◽  
pp. 488-495
Author(s):  
HOGER GHAHRAMANI
Keyword(s):  
Banach Space ◽  
Inverse Semigroup ◽  
Cohomology Group ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Mild Conditions ◽  
First Cohomology Group

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$-bimodule which is an $L$-embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.

scholarly journals The regular radical of semigroup rings of commutative semigroups

1992 ◽  
Vol 34 (2) ◽  
pp. 133-141 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Regular Semigroup ◽  
Commutative Semigroup ◽  
General Problem ◽  
Associative Ring ◽  
Complete Solution ◽  
Group Rings ◽  
Semigroup Rings ◽  
Regular Group ◽  
Commutative Semigroups

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

scholarly journals Moore-Penrose inversion in complex contracted inverse semigroup algebras

1999 ◽  
Vol 66 (3) ◽  
pp. 297-302
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Group Algebra ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Locally Finite ◽  
Complex Group ◽  
Complex Group Algebra

AbstractIt is shown that every element of the complex contracted semigroup algebra of an inverse semigroup S = S0 has a Moore-Penrose inverse, with respect to the natural involution, if and only if S is locally finite. In particular, every element of a complex group algebra has such an inverse if and only if the group is locally finite.

Trace functions on the algebras of certain E-unitary inverse semigroups

1995 ◽  
Vol 125 (5) ◽  
pp. 1077-1084 ◽  
Author(s):  
M. J. Crabb ◽  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Trace Function ◽  
Complex Field ◽  
Semigroup Algebras ◽  
Complex Conjugation ◽  
Arbitrary Rank

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

scholarly journals Module amenability of restricted semigroup algebras under module actions

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 787-793
Author(s):  
Abbas Sahleh ◽  
Somaye Tanha
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Clifford Semigroup ◽  
Module Amenability

In this article, weshow that module amenability with the canonical action of restricted semigroup algebra l1r (S) and semigroup algebra l1(Sr) are equivalent, where Sr is the restricted semigroup of associated to the inverse semigroup S. We use this to give a characterization of module amenability of restricted semigroup algebra l1r (S) with the canonical action, where S is a Clifford semigroup.

The Chain of Ideals of Regular Semigroup Algebras

2021 ◽  
Vol 11 (12) ◽  
pp. 2076-2086
Author(s):  
志荣 苏
Keyword(s):  
Regular Semigroup ◽  
Semigroup Algebras
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