Abundant semigroup algebras which are Azumaya

It is proved that any Azumaya semigroup algebra of an abundant semigroup over a field is a subdirect product of matrix algebras over Azumaya semigroup algebras of cancellative monoids. In particular, it is proved that for an abundant semigroup S with finitely many idempotents and a field K , K 0 [ S...

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Bibliographic Details
Published inSemigroup forum Vol. 103; no. 3; pp. 879 - 887
Main Authors Guo, Junying, Guo, Xiaojiang
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2021
Springer Nature B.V
Subjects
Algebra
Mathematics
Mathematics and Statistics
Monoids
Research Article
Semigroups
Semigroup algebra
Azumaya algebra
Abundant semigroup
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Summary:It is proved that any Azumaya semigroup algebra of an abundant semigroup over a field is a subdirect product of matrix algebras over Azumaya semigroup algebras of cancellative monoids. In particular, it is proved that for an abundant semigroup S with finitely many idempotents and a field K , K 0 [ S ] is Azumaya if and only if K 0 [ S ] is isomorphic to the direct sum of finite matrix algebras of Azumaya algebras of cancellative submonoids. This gives an affirmative answer to an open problem of Okniński in his monograph ( Semigroup Algebra , Marcel Dekker, New York, 1991) for the case that the semigroup is a finite abundant semigroup.
ISSN:0037-1912
1432-2137
DOI:10.1007/s00233-021-10214-w