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 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 outcome, and additionally, a similarity type [AJ within the Brauer workforce Br(k) of Azumaya algebras over a commutative ring okay includes all algebras B such that the corresponding different types mod-A and mod-B together with k-linear morphisms are identical by means of a k-linear functor. (For fields, Br(k) includes similarity periods of straightforward crucial algebras, and for arbitrary commutative ok, this is often subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) various different situations of a marriage of ring conception and type (albeit a shot­ gun wedding!) are inside the textual content. in addition, in. my try to extra simplify proofs, particularly to cast off 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 via the Morita context. As a spinoff, this offers beginning for a slightly entire thought of easy Noetherian rings-but extra approximately this within the introduction.

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Additional resources for Algebra: Rings, Modules and Categories I

Example text

A /I. b in A. The class of all lattices is closed and self-dual (see 10). 13. < Examples and Exercises. 13·1 Let A = Pow Y, where Y =1= 0. Then A is ordered by inclusion, and for a, b E A, a V b = a u b (set union) and a /I. b = a n b (set intersection). Hence, Pow Y is a lattice. Furthermore, for any set A, there is a duality d { Pow A -+ Pow A X~A -X. 2 Let 71, as usual, denote the set of integers. If a, b E 71, write a

An ordinal is countable if its cardinal is ~o. Thus, w is the least countable ordinal, that is, w = ~o. An ordinal ex is finite if ex E w. Thus, the finite ordinals are cardinals (integers). Since 1Pow wi> w = ~o, the question arises as to which ~~ is 1Pow w I. The continuum hypothesis states that c = 1 Pow wi = ~1' that is, that the cardinality of Pow w is the first uncountable cardinal. The generalized continuum hypothesis states that, for every ordinal ex, IPow~al = ~a+l' The set of real numbers on the closed interval [0, 1J is defined to be the continuum.

If A and B are well ordered sets, then there exists a unique initial order injection of one into the other. 23. Corollary (Comparability of Sets). If A and B are sets, then either there exists an injection A -lo- B or there exists an injection B ->- A. 2 A = B in case there is an order isomorphism A -lo- B; 23 -3 A < B in case A < B, but A =1= B. -lo- B; 24. Proposition. If A and B are well ordered sets, then precisely one of the following holds: A < B, A = B, or B < A. A set A is transitive in case vx, y, z E A (y E x & z E Y ~ z E x) .

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