The rank of a commutative cancellative semigroup

2005 ◽  
Vol 107 (1-2) ◽  
pp. 71-75 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Cancellative Semigroup

scholarly journals On the equivalence of cancellative extensions of commutative cancellative semigroups by groups

1971 ◽  
Vol 12 (2) ◽  
pp. 187-192
Author(s):  
Charles V. Heuer
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ideal Extension ◽  
Cancellative Semigroup ◽  
Finitely Generated ◽  
Necessary And Sufficient

In [1] D. W. Miller and the author established necessary and sufficient conditions for the existence of a cancellative (ideal) extension of a commutative cancellative semigroup by a cyclic group with zero. The purpose of this paper is to extend these results to cancellative extensions by any finitely generated Abelian group with zero and to establish in this general case conditions under which two such extensions are equivalent.

scholarly journals Cancellative semigroup rings which are Azumaya algebras

1988 ◽  
Vol 117 (2) ◽  
pp. 290-296 ◽  
Author(s):  
J Okninski ◽  
F Van Oystaeyen
Keyword(s):  
Cancellative Semigroup ◽  
Semigroup Rings ◽  
Azumaya Algebras

scholarly journals Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

scholarly journals Invariant vector means and complementability of Banach spaces in their second duals

2022 ◽  
Author(s):  
Radosław Łukasik
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Cancellative Semigroup ◽  
Invariant Mean ◽  
Invariant Vector ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractLet X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of $$X^{**}$$ X ∗ ∗ . We then show that X is complemented in $$X^{**}$$ X ∗ ∗ if and only if there exists an invariant mean $$M:\ell _\infty (S,X)\rightarrow X$$ M : ℓ ∞ ( S , X ) → X . This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).

scholarly journals Algebras of cancellative semigroups

1994 ◽  
Vol 49 (1) ◽  
pp. 165-170 ◽  
Author(s):  
Jan Okninski
Keyword(s):  
Jacobson Radical ◽  
Semigroup Ring ◽  
Cancellative Semigroup ◽  
Unique Product

The Jacobson radical J(K[S]) of the semigroup ring K[S] of a cancellative semigroup S over a field K is studied. We show that, if J(K[S]) ≠ 0, then either S is a reversive semigroup or K[S] has many nilpotents and J(K[P]) ≠ 0 for a reversive subsemigroup P of S. This is used to prove that J(K[S]) = 0 for every unique product. semigroup S.

scholarly journals Prime and semiprime semigroup rings of cancellative semigroups

1993 ◽  
Vol 35 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Group Ring ◽  
Cancellative Semigroup ◽  
Semigroup Rings ◽  
Similar Idea

Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K[S] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K[G], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G. In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S. He defined a subsemigroup Δ(S) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K[S]. His results were strengthened by the author in [7], where Δ(S) was also applied in the context of prime and semiprime algebras K[S]. However, Δ(S) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations.

On cancellative semigroup rings

1987 ◽  
Vol 15 (8) ◽  
pp. 1667-1677 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Cancellative Semigroup ◽  
Semigroup Rings

scholarly journals Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups

2014 ◽  
Vol 57 (2) ◽  
pp. 533-564 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Cancellative Semigroup ◽  
Left Inverse ◽  
C Algebras ◽  
Inverse Hull

AbstractTo each discrete left cancellative semigroup S one may associate an inverse semigroup Il(S), often called the left inverse hull of S. We show how the full and reduced C*-algebras of Il(S) are related to the full and reduced semigroup C*-algebras for S, recently introduced by Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.

scholarly journals The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

1985 ◽  
Vol 26 (2) ◽  
pp. 107-113 ◽  
Author(s):  
E. Jespers ◽  
J. Krempa ◽  
P. Wauters
Keyword(s):  
Quotient Group ◽  
Associative Ring ◽  
Semigroup Ring ◽  
Cancellative Semigroup ◽  
Semigroup Rings ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.

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