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 a straightforward final result, and furthermore, a similarity category [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok includes all algebras B such that the corresponding different types mod-A and mod-B including k-linear morphisms are an identical through a k-linear functor. (For fields, Br(k) involves similarity periods of easy significant algebras, and for arbitrary commutative ok, this is often subsumed less than the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) a number of different situations of a marriage of ring conception and class (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, particularly to do away with the necessity for tensor items in Bass's exposition, I exposed a vein of principles and new theorems mendacity wholely inside of ring conception. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the root for it's a corre spondence theorem for projective modules (Theorem four. 7) instructed by way of the Morita context. As a spinoff, this gives beginning for a slightly entire idea of straightforward Noetherian rings-but extra approximately this within the introduction.

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**Algebra: Rings, Modules and Categories I**

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 whilst there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward end result, and in addition, a similarity classification [AJ within the Brauer workforce 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 along with k-linear morphisms are identical by means of a k-linear functor.

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**Example text**

1 Prove that (A X B) X C ~ A X B X C and A X (B X C) ~ A X B X C for sets A, B, and C. Generalize. Compare A X B with the set of all ordered pairs (a, b), a E A, b E B. 2 A mapping t: X -+ Y is said to be epic in case it is true that for every pair of mappings gi: Y -+ Z, i = 1, 2, the following implication holds: gd = g2t::::;, gl = g2' Show that a mapping of sets is epic if and only if it is surjective. The mapping t: X -+ Y is said to be monic if it is true that for every pair of mappings hi: U -+ X, i = 1, 2, the following implication holds: t hl = th 2 ::::;, hI = h2 • Show that a mapping is monic if and only if it is injective.

Well Ordering Theorem The well ordering theorem states that any set A can be well ordered. This theorem is a controversial one in mathematics. The controversy is typified by the following question: IflR can be well ordered, what is the ordering? The point is that a well ordering of JR is difficult, if not impossible, to visualize. Expressed otherwise, there is no known effective procedure that will determine in a well ordering of 1R when a >b for any pair a, b E JR. For this reason, a minority of mathematicians prefer not to use this theorem.

4 For any sets A and B, show that the set of bijections of A is equivalent to the set of bijections of B if and only if A is equivalent to B. If A has infinite cardinality IX, show that the set of bijections of A has cardinality IX~. Assuming the generalized continuum hypothesis, prove that the set of bijections of A has cardinality 2~, where IX is the cardinality of A. 5 Prove that IXP, IX + p, and ",P are countable when IX and pare countable. 6 Show that cardinal sum and product are commutative, associative, and distributive (see Chapter 1 for definitions).