# 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 an easy outcome, and furthermore, a similarity type [AJ within the Brauer staff 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) comprises similarity periods of easy primary algebras, and for arbitrary commutative ok, this can be subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer staff. ) various different situations of a marriage of ring concept and type (albeit a shot­ gun wedding!) are inside the textual content. moreover, in. my try and additional simplify proofs, particularly to cast off the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside ring idea. 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 gives beginning for a slightly whole conception of easy Noetherian rings-but extra approximately this within the introduction.

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Now X E Pow A, and, since I is surjective, there exists a E A such that I (a) = X. If a EX, then by the definition of X, a ~ I (a) = X. This contradiction forces a to lie outside of X. But, again by the definition of X, a E I(a) = X. In summary, a E X and a ~ X, a contradiction that proves Cantor's theorem. Russell's Paradox There are sets that are not members of themselves. Let A denote the set of all those sets that are not members (or elements) of themselves; that is, A = (B IB is a set and B ~ B).

From the definition of A we deduce that A ~ A, a contradiction. Then certainly A ~ A is the case. But if this is true, then the definition of A implies that A E A, another contradiction. This example of a "set", and the contradiction it causes, is known as Russell's paradox (after its discoverer, Bertrand Russell). Cantor's Paradox Cantor's theorem leads to the following statement. 6. Cantor's Paradox. There does not exist a set A having the property that lor any set B there exists an infective mapping I: B ~ A .

This is the relation in B corresponding to the subset fllln (B X B) of B X B. One of the most general concepts in mathematics is that of an equivalence relation. An equivalence relation in a set A is a binary relation flll in A satisfying the three conditions for any a,b,c E A: Reflexivity: a flll a. Transitivity: If aflll band bflll c, then a:71 c. Symmetry: If aflllb, then bfllla. If flll is an equivalence relation in A, then for aEA, fllla= IbEA Jbfllla} is called the equivalence class determined by (or containing) a.