Algebra: Rings, Modules and Categories I by Carl Faith

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 resolution organizes rules so successfully that the classical Wedderburn-Artin theorem is a straightforward final result, and in addition, a similarity classification [AJ within the Brauer crew 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 together with k-linear morphisms are similar by way of a k-linear functor. (For fields, Br(k) involves 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 workforce. ) a variety of different situations of a marriage of ring concept and class (albeit a shot­ gun wedding!) are inside the textual content. moreover, in. my try and additional simplify proofs, particularly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside ring concept. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre­ spondence theorem for projective modules (Theorem four. 7) steered by means of the Morita context. As a derivative, this gives beginning for a slightly whole concept of easy Noetherian rings-but extra approximately this within the introduction.

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A set A is transitive in case vx, y, z E A (y E x & z E Y ~ z E x) . ly~ (y E x) V (y = x). Thus, if A is transitive, then and y < x ~ (y < x) V (y = x). In other words, if A is transitive, then the E-relation orders A. An ordinal is defined to be a transitive set in which the E-relation is a well ordering. 25. Proposition. The following properties of ordinals hold: Any initial segment of an ordinal is an ordinal. 2 If ex is an ordinal, and if x E ex, then x = (y E ex I y < x), and x is an ordinal.

A is said to have a least (resp. greatest) element if there is an element a E A satisfying a

8 For any infinite cardinal number a and natural number n, an = a and na=a. 9 Let X and Y be infinite sets, and let For each y E Y, let ay = 1/- 1 (y) I. - Y be a surjeetion. I: LEY ay) . 1 Y 1 . 10 Start with a countable sequence {F" 1 n = 1,2, ... ¥. ¥, fP) is a finite or infinite sequence of elements bl , ... , b", ... ¥ such that bi E Fi and bi+! E fP (b i ), i = 1, 2, .... The length of a finite path bl , ••• , bm is m; the length of the infinite path bl , b2 , ••• is infinite. Prove: Konig's Graph Lemma.

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