Get Algebra Rings Modules and Categories I (Grundlehren der mathematischen Wissenschaften)
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Book Details :
Published on: 2012-03-08
Released on: 2012-03-08
Original language: English
VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction. Ring (mathematics) - Wikipedia In mathematics a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that ... Category theory - Wikipedia Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic ... [ihtik.lib.ru] _ [ihtik.lib.ru] _. : 14292 : 573 GB; ; d:\_ihtik.lib.ru\2012.03 ... nesinkoyleri.org 1 1958 1965 1. 2 1957 1. 3 1956 2005 2. 4 2007 2007 2. 5 1995 1995 1. 6 2011 1. 7 1958 1. 8 1. 9 1999 1999 1. 10 1991 1991 1. 11 2001 2002 1. 12 1972 2010 2. 13 1996 ...
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