# 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.

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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

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.