By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "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 jewelry A and B. Morita's resolution organizes rules so successfully that the classical Wedderburn-Artin theorem is an easy outcome, and in addition, a similarity category [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring okay contains all algebras B such that the corresponding different types mod-A and mod-B inclusive of k-linear morphisms are similar by way of a k-linear functor. (For fields, Br(k) comprises similarity periods of easy important algebras, and for arbitrary commutative ok, this is often subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer workforce. ) quite a few different cases of a marriage of ring idea and class (albeit a shot gun wedding!) are inside the textual content. in addition, in. my try to 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 idea. 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) steered via the Morita context. As a spinoff, this gives beginning for a slightly whole concept of easy Noetherian rings-but extra approximately this within the introduction.

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**Example text**

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.

2. If g: B -+ A is as given, and if b E B, then b = 1B(b) = f(g(b)) E im/, thus I is surjective. ) For each bE B, let Ab = {a E A I I(a) = b}. Each set Ab is nonempty, and in each Ab choose one element b' . Thus, there is a mapping g: B -+ A, where g(b) = b' V bE B. Clearly, Ig = 1B. 3. By 1. , f is bijective if and only if there exist mappings g: B -+ A and h: A -)- B such that gf = 1A and Ih = 1R • 16 Foreword on Set Theory In this case, V b E B, vve have g(b) = g(th(b)) = g/(h(b)) = h(b), consequently, g = h.

26-29J, or Halmos [60, p. ]. ) 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.