Download Algebra: Rings, Modules and Categories I by Carl Faith 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 while there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's resolution organizes rules so successfully that the classical Wedderburn-Artin theorem is a straightforward outcome, and additionally, a similarity type [AJ within the Brauer staff 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 which include k-linear morphisms are identical through a k-linear functor. (For fields, Br(k) comprises similarity sessions of straightforward primary algebras, and for arbitrary commutative okay, this can be subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) a variety of different cases of a marriage of ring conception and type (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try to additional simplify proofs, significantly to put off the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside ring concept. 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) urged by means of the Morita context. As a spinoff, this offers origin for a slightly entire concept of easy Noetherian rings-but extra approximately this within the introduction.

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Additional info for Algebra: Rings, Modules and Categories I

Example text

If flll is an equivalence relation in A, then for aEA, fllla= IbEA Jbfllla} is called the equivalence class determined by (or containing) a. The class of all equivalence classes of A under flll is a set denoted by flllA' and there is a surjective mapping called canonical. Conversely, any mapping of A determines an equivalence relation as in the following statement. 8. Statement. If R: A -+ X is a mapping, then there is an equivalence relation flll in A defined by b flll a # R(a) = R(b). The mapping is infective and is said to be induced by R, and there is a commutative diagram: R Ordered Sets 2t It Y is a set such that commutes, then there exists a unique equivalence Y -+ f7lA such that / y A~~ commutes.

We (partially) order TV by the rule: If (X, >1) and (Y, >2) are well ordered subsets of A, then (X, >1) > (Y, >2) if and only if: (1) X-=> Y; (2) >1 induces >2; and (3) if x E X and y E Y, and if y >1 x, then x E Y. TV is nonempty since TV contains the empty set. If C = {(Xi, > i) liE I} is any chain in TV, then (X, E TV, where X = UiEIXi and > is an ordering of X that induces >1 ViE I. Specifically, if x, y EX, then x, y E Xi for some f E I, and we define x> y if and only if x>iY' If also x, y EX", then x >i Y if and only if x >" y.

The proposition on the existence of coproducts (Proposition 19) states that an); two coproducts for the family IIi: Ai -+ Al are equivalent. ' In addition to the duality principle for ordered sets, there is a duality principle for mappings. 6) and will not be given here. Intuitively, this principle can be applied to any statement about mappings of sets to derive another statement by interchanging domain and range and reversing arrows. In this sense coproduct is dual to product, and surjection is dual to injection.

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