By Carl Faith
VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating principles 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 answer organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward end result, and in addition, a similarity category [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok contains all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are similar by means of a k-linear functor. (For fields, Br(k) contains similarity periods of easy crucial algebras, and for arbitrary commutative okay, this can be subsumed lower than the Azumaya 1 and Auslander-Goldman [60J Brauer team. ) various different cases of a marriage of ring idea and type (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, significantly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside of 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) instructed by means of the Morita context. As a spinoff, this offers starting place for a slightly whole conception of straightforward Noetherian rings-but extra approximately this within the introduction.
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Extra resources for Algebra: Rings, Modules and Categories I
This order is called reverse, or dual, order. The ordered set (A, >*) is called the ordered set dual to A and is denoted by A *. Thus, the identity mapping 1A : A -+ A is a duality A -+ A *. The very simple fact of the existence of A * for each ordered set A has useful consequences which seem to belie the triviality of this fact. The most important of these is the duality principle which asserts that for every theorem about ordered sets there is a dual theorem obtained simply by reversing order.
2 Let 71, as usual, denote the set of integers. If a, b E 71, write a b or b > a. In the former case a = a V b and b = a /I. b. 4 Any "geometric" lattice is a lattice. By geometric lattice we mean a configuration, oriented as in the diagram, consisting of two classes of parallel lines that intersect each other.
If A and B are ordered sets, then a mapping I: A -'J>- B is an order homomorphism provided that a>b§t(a»t(b). 'if a, b EA. Then t is an order injection (resp. surjection, bijection) in case t is an order homomorphism and an injective (resp. surjective, bijective) mapping. An order isomorphism I: A -'J>- B is a bijection such that both I and 1-1 are order homomorphisms. Note that every order injection I: A ->- B induces an order isomorphism I: A -'J>- im t, hence, any bijective order homomorphism is an order isomorphism.