On a characterization of Azumaya algebras
In his paper [J. Algebra 77, 323-332 (1982; Zbl 0491.16009)] \textit{A. Braun} proves the following theorem: Let R be a ring with 1 and centre Z(R). Then the following are equivalent: i) R is Azumaya. ii) There exists \(u\in R\otimes R^{op}\) such that u*R\(\subseteq Z(R)\) and \(u*1=1\), where * denotes the usual left \(R\otimes R^{op}\) module structure on R. The proof uses a generalization of a theorem of M. Artin, namely that if \(R=S\{x_ 1,...,x_ k\}\) is a p.i. ring, S a central noetherian subring, then R is Azumaya iff for every two sided ideal I in R, \(Z(R/I)=Z(R)/I\cap Z(R)\). This note gives a short direct proof that ii)\(\Rightarrow i)\).
scientific article; zbMATH DE number 4081754