On semigroup algebras of cancellative commutative semigroups

1991 ◽  
Vol 118 (1-2) ◽  
pp. 1-3 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Commutative Semigroup ◽  
Semigroup Algebras ◽  
Commutative Semigroups ◽  
Hereditary Radical

SynopsisA cancellative commutative semigroup s and a hereditary radical ρ are constructed such that ρ is S-homogeneous but not S-normal. This answers a question which arose in the literature.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

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.

Bases for commutative semigroups and groups

2008 ◽  
Vol 145 (3) ◽  
pp. 579-586 ◽  
Author(s):  
NEIL HINDMAN ◽  
DONA STRAUSS
Keyword(s):  
Abelian Group ◽  
Finite Order ◽  
Commutative Semigroup ◽  
Finite Subset ◽  
Surprising Result ◽  
Commutative Semigroups ◽  
Or Groups

AbstractA base for a commutative semigroup (S, +) is an indexed set 〈xt〉t∈A in S such that each element x ∈ S is uniquely representable as Σt∈Fxt where F is a finite subset of A and, if S has an identity 0, then 0 = Σn∈Øxt. We investigate those commutative semigroups or groups which have a base. We obtain the surprising result that has a base. More generally, we show that an abelian group has a base if and only if it has no elements of odd finite order.

On homogeneous radicals of semigroup rings of commutative semigroups

1993 ◽  
Vol 123 (5) ◽  
pp. 951-957 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Semigroup Rings ◽  
Commutative Semigroups ◽  
Hereditary Radical

SynopsisAll Archimedean commutative semigroups S are described such that every S-homogeneous hereditary radical is S-normal. It is shown that this result is in a sense unimprovable.

On commutative semigroup algebras

1983 ◽  
Vol 93 (2) ◽  
pp. 237-246 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Algebras ◽  
Necessary And Sufficient

This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).

On the Jacobson radical of semigroup rings of commutative semigroups

1990 ◽  
Vol 108 (3) ◽  
pp. 429-433 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Associative Ring ◽  
Complete Solution ◽  
Semigroup Rings ◽  
Commutative Semigroups ◽  
Free Commutative Semigroup

Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S, the equalities B(RS) = B(R)S and L(RS) = L(R)S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π(RS) = π(R)S, where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].

On the Jacobson radical of certain commutative semigroup algebras

1984 ◽  
Vol 96 (1) ◽  
pp. 15-23 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Commutative Ring ◽  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Semigroup Algebras

In two previous papers the author studied the Jacobson and nil redicals of the algebra of a commutative semigroup over a field [8] and over a commutative ring with unity [9]. This work is continued here.

scholarly journals Solution of Several Functional Equations on Nonunital Semigroups Using Wilson’s Functional Equations with Involution

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jaeyoung Chung ◽  
Prasanna K. Sahoo
Keyword(s):  
Commutative Semigroup ◽  
Functional Equations ◽  
Complex Numbers ◽  
Commutative Semigroups ◽  
General Solutions

LetSbe a nonunital commutative semigroup,σ:S→San involution, andCthe set of complex numbers. In this paper, first we determine the general solutionsf,g:S→Cof Wilson’s generalizations of d’Alembert’s functional equations  fx+y+fx+σy=2f(x)g(y)andfx+y+fx+σy=2g(x)f(y)on nonunital commutative semigroups, and then using the solutions of these equations we solve a number of other functional equations on more general domains.

scholarly journals On group graded rings satisfying polynomial identities

1995 ◽  
Vol 37 (2) ◽  
pp. 205-210 ◽  
Author(s):  
A. V. Kelarev ◽  
J. Okniński
Keyword(s):  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Normal Band ◽  
Ring Theory ◽  
Semigroup Ring ◽  
Semigroup Algebras ◽  
Graded Rings ◽  
Finitely Generated ◽  
Polynomial Identities ◽  
Locally Finite

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

scholarly journals A characterization of Commutative Semigroups

2021 ◽  
Vol 12 (3) ◽  
pp. 5150-5155
Author(s):  
Dr. D. Mrudula Devi Et. al.
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Commutative Semigroup ◽  
Zero Semigroup ◽  
Completely Regular Semigroup ◽  
Semi Group ◽  
Commutative Semigroups ◽  
Completely Regular ◽  
Left Regular

This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup  is u – inverse semigroup. We will also prove that if (S,.) is a H -  semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup  and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

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