Download E-books Algebra: Rings, Modules and Categories I (Grundlehren der mathematischen Wissenschaften) PDF

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 whilst there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy final result, and additionally, a similarity type [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring okay comprises all algebras B such that the corresponding different types mod-A and mod-B along with k-linear morphisms are an identical by means of a k-linear functor. (For fields, Br(k) comprises similarity sessions of straightforward important algebras, and for arbitrary commutative ok, this is often subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer team. ) quite a few different situations of a marriage of ring conception and type (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try to extra simplify proofs, particularly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside ring thought. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the root for it's a corre­ spondence theorem for projective modules (Theorem four. 7) steered through the Morita context. As a spinoff, this gives beginning for a slightly entire idea of easy Noetherian rings-but extra approximately this within the advent.

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