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 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.

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 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.

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.

scholarly journals Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad ◽  
T. Srinivas
Keyword(s):  
Jacobson Radical ◽  
Paper Properties ◽  
Hereditary Radical ◽  
Constant Part ◽  
The Right

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

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