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 whilst there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy end result, and in addition, a similarity classification [AJ within the Brauer team 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 which include k-linear morphisms are identical by means of a k-linear functor. (For fields, Br(k) includes similarity sessions of straightforward important algebras, and for arbitrary commutative okay, this is often subsumed less than the Azumaya 1 and Auslander-Goldman [60J Brauer team. ) a number of different cases of a marriage of ring conception and type (albeit a shot gun wedding!) are inside the textual content. moreover, in. my try and additional simplify proofs, significantly to cast off the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside of 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 by means of the Morita context. As a spinoff, this offers origin for a slightly entire concept of straightforward Noetherian rings-but extra approximately this within the introduction.
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Extra resources for Algebra: Rings, Modules and Categories I
3 I is bifective il and only 1:1 I has an inverse. ---7- B be a mapping 01 nonempty sets. Then The proposition establishes that the set of bijections X ---7- X of a non empty set X is an example of a group (defined in Chapter 1) with respect to the operation of composition of mappings. If X is a set containing not less than three elements, then this group is noncommutative in the sense that there exist I and g (bijections) such that I g =1= g I. ) Equivalent Sets Two sets A and B are equivalent if there exist mappings f: A ->- B and g: B ---7-A such that gl = 1A and Ig = 1B • In this case, we write A ~ B; otherwise, A and B are said to be inequivalent.
Is reserved to mean "unique". ' Axiom of Replacement Vt l , ... y(A n (x, y; tv B (u, v) ... , t k ) =? Vu3vB(u, = Vr(r E v ~ 3s(s Eu & An (s, r; t l , ... , v))), where tk)))' This axiom can be phrased in the following way. If for fixed t l , ... , it is true that An (x, y; fl' ... , tk ) defines y uniquely as a function of x, say y = 1(x), then for each u, the image of 1 on u is a set. (The role of the tv ... ) ZF6n is a powerful axiom in that it allows forming a new set by putting together sets chosen from a universe of sets (defined directly).
Is A E A? Suppose so. 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 .
Algebra: Rings, Modules and Categories I by Carl Faith