## Algebra: Rings, Modules and Categories I by Carl Faith

By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating rules of Chase and Schanuel. one of many Morita theorems characterizes while there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's answer organizes rules so successfully that the classical Wedderburn-Artin theorem is an easy end result, and in addition, a similarity category [AJ within the Brauer team Br(k) of Azumaya algebras over a commutative ring ok comprises all algebras B such that the corresponding different types mod-A and mod-B inclusive of k-linear morphisms are similar by means of a k-linear functor. (For fields, Br(k) includes similarity sessions of straightforward critical algebras, and for arbitrary commutative ok, this is often subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer workforce. ) various different cases of a marriage of ring conception and type (albeit a shot gun wedding!) are inside the textual content. in addition, in. my try and extra simplify proofs, significantly to get rid of the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside ring concept. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre spondence theorem for projective modules (Theorem four. 7) advised via the Morita context. As a spinoff, this offers beginning for a slightly whole conception of straightforward Noetherian rings-but extra approximately this within the introduction.