Azumaya rings with locally perfect centers
An Azumaya ring is a ring R in which i) every ideal is generated by central elements, ii) every ring homomorphism from R maps the centre of R onto the centre of the image of R. If R is an Azumaya ring with centre Z then R/RM is a central simple Z/M-algebra for every maximal ideal M of Z. For the converse, the classical known results show that this is true for rings which are module finite over their centre. In this paper the author exhibits that the converse is true whenever Z is locally perfect. The author sharpens the condition on T by showing that if T is a locally perfect subring of the centre of R, such that R/RM is a central simple T/M-algebra for every maximal ideal M of T, then R is an Azumaya ring with centre T. This allows the author to exhibit that the central ideal algebras over perfect rings due to \textit{M. L. Ranga Rao} [Bull. Am. Math. Soc. 78, 588-592 (1972; Zbl 0258.16004)] are indeed Azumaya rings.
scientific article; zbMATH DE number 4055803
scientific article