On Artin's theorem and Azumaya algebras

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A short proof of the existence of central polynomial of matrix rings is given and some of its applications. The following characterization of Azumaya algebras is proved: R is Azumaya iff there exist ai, bi ε{lunate} R, i = 1,..., k with ∑ik = 1 airbi ε{lunate} Z(R), ∀r ε{lunate} R and ∑ aibi = 1. This is proved as a consequence of the following generalization of a theorem due to M. Artin (and generalized by C. Procesi): Let R = Λ{x1,..., xk} be a p.i. ring, Λ a central noetherian subring. Then R is Azumaya iff for every two sided ideal I in R, Z( R I) = Z(R) I ∩ Z(R), where Z(R) denotes the center of R.

Original languageEnglish
Pages (from-to)323-332
Number of pages10
JournalJournal of Algebra
Issue number2
StatePublished - Aug 1982

ASJC Scopus subject areas

  • Algebra and Number Theory


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