By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating principles 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 jewelry A and B. Morita's resolution organizes rules so successfully that the classical Wedderburn-Artin theorem is an easy end result, and furthermore, a similarity category [AJ within the Brauer staff Br(k) of Azumaya algebras over a commutative ring okay includes all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are similar by way of a k-linear functor. (For fields, Br(k) comprises similarity sessions of straightforward valuable algebras, and for arbitrary commutative okay, this is often subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer workforce. ) a number of different situations of a marriage of ring idea and classification (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and extra simplify proofs, particularly to put off the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside ring idea. 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) prompt by means of the Morita context. As a derivative, this gives beginning for a slightly entire idea of easy Noetherian rings-but extra approximately this within the introduction.

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The well ordering theorem discussed later). » 11. » Exercises. 1 If (A, is an ordered set, and if every non-empty subset has a least element, then A is a well ordered set. 2 If A and B are well ordered sets, show that A u B and A X B can be well ordered in a "natural" way. Minimum and Maximum Conditions An ordered set A satisfies the maximum (resp. minimum) condition in case every nonempty subset X of A contains an element that is maximal (resp. minimal) in X. Any well ordered set A satisfies the minimum condition, and A in its reverse order satisfies the maximum condition.

For fixed and if {J is a limit ordinal, iX P = iX, iX' iX • 0 = 0 and IX' ({3 + 1) {3 = sup (iX • Y I y < {J}. define iXO = 1, iXP+l = sup {iX~ I y IX~ • iX, < {3} . Cardinals Let A and B be sets. Write IA I < IBI, if there exists an injection A ---+ B, and write IA I = IB I in case there is a bij ection A ---+ B. An earlier theorem implies that if A and B are sets, then either IA I < IB I or else IB I < IA I. The Cantor-Schrader-Bernstein theorem asserts the implication: IAI < IBI & IBI < IAI~ IAI = IBI· We say that A and B have the same cardinality (numerosity) when IAI= IBI· A cardinal is defined to be an ordinal implication holds for ordinals {3: {J < iX~ I{JI < iX such that the following liXl· This defines a cardinal to be the least ordinal of a given cardinality.

Exercises. 1 If (A, is an ordered set, and if every non-empty subset has a least element, then A is a well ordered set. 2 If A and B are well ordered sets, show that A u B and A X B can be well ordered in a "natural" way. Minimum and Maximum Conditions An ordered set A satisfies the maximum (resp. minimum) condition in case every nonempty subset X of A contains an element that is maximal (resp. minimal) in X. Any well ordered set A satisfies the minimum condition, and A in its reverse order satisfies the maximum condition.