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 resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward final result, and in addition, a similarity category [AJ within the Brauer workforce Br(k) of Azumaya algebras over a commutative ring ok involves all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are identical by way of a k-linear functor. (For fields, Br(k) comprises similarity sessions of straightforward important algebras, and for arbitrary commutative ok, this can be subsumed lower than the Azumaya 1 and Auslander-Goldman [60J Brauer staff. ) various different situations of a marriage of ring conception and classification (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, particularly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of principles 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 by means of the Morita context. As a derivative, this offers starting place for a slightly whole thought of straightforward Noetherian rings-but extra approximately this within the introduction.
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Additional info for Algebra: Rings, Modules and Categories I
2. The axioms of ZF are infinite in number because of the axiom of replacement; however, the GB axioms are finite in number (d. Cohen [66, pp. 13 and 83]). 3. G6del  proved that the axiom of choice (AC) (and also the generalized continuum hypothesis (GCH), which is stated following Theorem 39 in the Foreword) is consistent with the GB axioms for ~~. \ 4. Cohen [63, 64] proved that the axiom of choice (also the GCH) is independent of the GB axioms. 5. Sierpinski  proved that GCH implies AC.
Then C* is called the class dual to C. Since C is closed under order isomorphism, and since A is order isomorpliic to (A *)* for every A E C, it follows that C = C**, where X** denotes (X*)*. We say that C is self-dual provided that C = C*. Thus, C is self-dual whenever, V A E C, it is true that A E C implies A * E C. 24 10. Foreword on Set Theory Duality Principle. 1 If 5 is a theorem abottt ordered sets, then 5* is a theorem about ordered sets, called the theorem dual to 5. 2 Let C be a closed class of ordered sets axiomatized by a set 5 of statements.
B) There are five nonisomorphic ordered sets of three elements, three of which are self-dual. (c) The number of nonisomorphic ordered sets of four elements is 16, but there are 219 different orders possible (d. Birkhoff [48, p. 4J). 12 three of which are self-dual. (b) There are just four nonisomorphic lattices of less than five elements. (c) There are just 15 lattices of six elements, of which exactly seven are self-dual (d. Birkhoff [48, p. 17J). 13 contains a sub lattice of exactly six elements.