Cancellative semigroup rings which are Azumaya algebras

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.

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Prime and semiprime semigroup rings of cancellative semigroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500009514 ◽

1993 ◽

Vol 35 (1) ◽

pp. 1-12 ◽

Cited By ~ 3

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.

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On cancellative semigroup rings

Communications in Algebra ◽

10.1080/00927878708823495 ◽

1987 ◽

Vol 15 (8) ◽

pp. 1667-1677 ◽

Cited By ~ 4

Author(s):

Jan Okniński

Keyword(s):

Cancellative Semigroup ◽

Semigroup Rings

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The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500005863 ◽

1985 ◽

Vol 26 (2) ◽

pp. 107-113 ◽

Cited By ~ 14

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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Canonical trace ideal and residue for numerical semigroup rings

Semigroup Forum ◽

10.1007/s00233-021-10205-x ◽

2021 ◽

Author(s):

Jürgen Herzog ◽

Takayuki Hibi ◽

Dumitru I. Stamate

Keyword(s):

Numerical Semigroup ◽

Semigroup Rings ◽

Trace Ideal ◽

Numerical Semigroup Rings

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The resolution property via Azumaya algebras

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0002 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Siddharth Mathur

Keyword(s):

Azumaya Algebras ◽

Algebraic Space ◽

Local Methods ◽

Artin Stacks ◽

Resolution Property

Abstract Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.

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