Carl Faith's Algebra: Rings, Modules and Categories I PDF

By Carl Faith

ISBN-10: 3642806341

ISBN-13: 9783642806346

ISBN-10: 3642806368

ISBN-13: 9783642806360

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "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 rules so successfully that the classical Wedderburn-Artin theorem is an easy outcome, and in addition, a similarity category [AJ within the Brauer workforce Br(k) of Azumaya algebras over a commutative ring okay comprises all algebras B such that the corresponding different types mod-A and mod-B including k-linear morphisms are similar by way of a k-linear functor. (For fields, Br(k) contains similarity sessions of easy primary algebras, and for arbitrary commutative ok, this can be subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) various different situations of a marriage of ring idea and classification (albeit a shot­ gun wedding!) are inside the textual content. moreover, in. my try to extra simplify proofs, significantly to get rid of the necessity for tensor items in Bass's exposition, I exposed a vein of rules 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 foundation 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 origin for a slightly whole thought of easy Noetherian rings-but extra approximately this within the introduction.

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Extra info for Algebra: Rings, Modules and Categories I

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We call the reverse order exclusion. Then, there is a duality d{U-+U X~U-X which maps intersections onto unions and unions onto intersections. Thus, for any family {Xi l iE1 of sets of U, U- and UXi = iEI n (U - iEI Xi) For any statement S about ordered sets, there is a dual statement S* obtained from S by reversing order, that is, by interchanging > and < throughout. The statement S is self-dual provided that S = S*. A set {Silw of statements about ordered sets is self-dual provided that there exists a permutation p of I such that S; = Sp(i) ViE I.

Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publ. 1942. Cohen, P. : Set Theory and the Continuum Hypothesis. New York: Benjamin 1966. - The independence of the continuum hypothesis, I, II. Proc. Nat. Acad. Sci. A. 50, 1143-1148 (1963); 51,105-110 (1964). Cohn, P. : Universal Algebra. New York: Harper & Row 1965. : Uber den Degriff "deiinit" und die Unabhangigkeit des Auswahl axioms. Sitzungsber. Preuss. Akad. wiss. -math. 1922, 253-257. : Set Theory. Amsterdam: North-Holland Publ.

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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Algebra: Rings, Modules and Categories I by Carl Faith

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