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 few "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 jewelry A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy end result, and in addition, a similarity category [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 including k-linear morphisms are similar by way of a k-linear functor. (For fields, Br(k) comprises similarity sessions of easy relevant algebras, and for arbitrary commutative ok, this can be subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer staff. ) various different situations of a marriage of ring thought and classification (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try and extra simplify proofs, particularly to do away with the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside of ring conception. 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) advised by way of the Morita context. As a derivative, this gives starting place for a slightly entire thought of easy Noetherian rings-but extra approximately this within the advent.

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