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].

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.

Radicals of semigroup rings of commutative semigroups

1986 ◽  
Vol 99 (3) ◽  
pp. 435-445 ◽  
Author(s):  
J. Okniński ◽  
P. Wauters
Keyword(s):  
Commutative Ring ◽  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Semigroup Rings ◽  
Commutative Semigroups

In this paper we determine radicals of semigroup rings R[S] where R is an associative, not necessarily commutative, ring and S is a commutative semigroup. We will restrict ourselves to the prime radical P, the Levitzki radical L and the Jacobson radical J. At the end we will also give a few comments on the Brown-McCoy radical U.

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.

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.

scholarly journals On p-injective rings

1995 ◽  
Vol 37 (3) ◽  
pp. 373-378 ◽  
Author(s):  
Gennadi Puninski ◽  
Robert Wisbauer ◽  
Mohamed Yousif
Keyword(s):  
Direct Summand ◽  
Jacobson Radical ◽  
Associative Ring ◽  
The Right ◽  
Left Annihilator

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

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.

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