Local-global results for modules over algebras and Azumaya rings
For any \(R\)-module \(M\), let \(\sigma[M]\) be the category of \(M\)-generated modules. The author calls a module \(M\) such that the functor \(\hbox{Hom}(M,-): \sigma[M]\to \hbox{End}(M)\hbox{-Mod}\) is an equivalence of categories a self-progenerator. Let \(Z\) be a commutative ring. A not necessarily associative algebra \(A\) can be viewed as a module over its multiplication algebra \(M(A)\). \(A\) is an Azumaya ring if it is a self-progenerator as \(M(A)\)-module, and an Azumaya algebra if it is a self-progenerator for all \(M(A)\)-modules. The author obtains local-global characterizations for Azumaya rings (Corollary 3.3), Azumaya algebras (Corollary 3.5) and biregular rings (Corollary 3.7, 3.8). Localization is taken with respect to maximal ideals of \(Z\) and with respect to the maximal ideals in the ring of idempotents of \(Z\) (Pierce stalks). The results include and improve corresponding results for associative rings by Burkholder, Dauns and Hofmann, Szeto and others.
scientific article; zbMATH DE number 7134
scientific article