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 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 earrings A and B. Morita's answer organizes rules so successfully that the classical Wedderburn-Artin theorem is an easy end result, and in addition, a similarity category [AJ within the Brauer team 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 inclusive of k-linear morphisms are similar by means of a k-linear functor. (For fields, Br(k) includes similarity sessions of straightforward critical algebras, and for arbitrary commutative ok, this is often subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer workforce. ) various different cases of a marriage of ring conception and type (albeit a shot­ gun wedding!) are inside the textual content. in addition, in. my try and 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 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) advised via the Morita context. As a spinoff, this offers beginning for a slightly whole conception of straightforward Noetherian rings-but extra approximately this within the introduction.

Show description

Read or Download Algebra: Rings, Modules and Categories I PDF

Similar algebra & trigonometry books

Spectral theory of automorphic functions

Venkov A. B. Spectral conception of automorphic features (AMS, 1983)(ISBN 0821830783)

Diskrete Mathematik fuer Einsteiger

Dieses Buch eignet sich hervorragend zur selbstständigen Einarbeitung in die Diskrete Mathematik, aber auch als Begleitlektüre zu einer einführenden Vorlesung. Die Diskrete Mathematik ist ein junges Gebiet der Mathematik, das eine Brücke schlägt zwischen Grundlagenfragen und konkreten Anwendungen. Zu den Gebieten der Diskreten Mathematik gehören Codierungstheorie, Kryptographie, Graphentheorie und Netzwerke.

Structure of algebras,

The 1st 3 chapters of this paintings comprise an exposition of the Wedderburn constitution theorems. bankruptcy IV comprises the speculation of the commutator subalgebra of an easy subalgebra of a typical uncomplicated algebra, the examine of automorphisms of an easy algebra, splitting fields, and the index aid issue concept.

Additional resources for Algebra: Rings, Modules and Categories I

Sample text

Let a be a least element of Y n Xi with respect to the ordering >i. If bEY, then bEY n Xi for some f. Suppose b' of X' as follows: If x, y E X', and if x = a, then x >' y; the remaining possibility is when both x, y EX, in which case let x >' y if and only if x > y.

Provided the following holds: If a l < a2 < ... < an < ... (resp. a l > a2 > ... > an > ... ) holds for any countable sequence {an In = 1,2, ... J of elements of A, then there exists an integer k such that an = ak V n > k. 12. Proposition. In an ordered setA, the maximum (resp. minimum) condition is equivalent to the ascending (resp. descending) chain condition. Proof. Assume the maximum condition, and let a l < a2< ... < an < ... be a countable chain of elements of A. Let a be a maximal element in the subset {anln = 1,2, ...

Thus, 1x (x) = x V x EX. The mappmg -7 X induced by the identity mapping 1x: X -7 X is called the inclusion mapping of U in X. We also denote the inclusion by '1 f} . Note that any mapping h: U -7 Y can be extended to a mapping of f : X -7 Y; for example, pick any element Y E Y, and choose f so that f I (X - U) = Yeanst. In general, h may be extended in many ways. If f: A -7 B is a mapping, then im f is a subset of B consisting of all bE B for which b = f(a) for some a E A. If imf =F B, then there is another function h: A -7 im f defined by h(a) = f(a) V a EA.

Download PDF sample

Rated 4.88 of 5 – based on 26 votes