Read e-book online Algebra: Rings, Modules and Categories I PDF

By Carl Faith

ISBN-10: 3642806341

ISBN-13: 9783642806346

ISBN-10: 3642806368

ISBN-13: 9783642806360

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 whilst 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 classification [AJ within the Brauer workforce 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 along with k-linear morphisms are identical through a k-linear functor. (For fields, Br(k) includes similarity sessions of easy imperative algebras, and for arbitrary commutative okay, this can be subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer team. ) a variety of different circumstances of a marriage of ring thought and class (albeit a shot­ gun wedding!) are inside the textual content. in addition, in. my try to extra simplify proofs, significantly to put off 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) prompt by means of the Morita context. As a spinoff, this offers beginning for a slightly whole idea of easy Noetherian rings-but extra approximately this within the introduction.

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Additional info for Algebra: Rings, Modules and Categories I

Example text

E. ]. Brouwer the author of a fix-point theorem in topology, which is proved using the axiom of choice! 15. Well Ordering Theorem. Any set A can be well ordered. Proof. The proof uses Zorn's lemma. Let TV denote the collection of all well ordered subsets of A. We (partially) order TV by the rule: If (X, >1) and (Y, >2) are well ordered subsets of A, then (X, >1) > (Y, >2) if and only if: (1) X-=> Y; (2) >1 induces >2; and (3) if x E X and y E Y, and if y >1 x, then x E Y. TV is nonempty since TV contains the empty set.

And h: W -7- Yare mappings. The diagram commutes, or is said to be commutative, in case kg = f. An arbitrary (polygonal) diagram commutes if each triangular sector commutes. Completed Diagrams Statements I and II are defined to have the same meaning: I. The diagram can be completed (by h: W -7- Y). Composition and Restriction 13 II. There exists h: W -+ Y and a commutative diagram X f • Y _ _ _ _ _--1... _ Y \/ w Similarly, to say that a diagram w -------h I, U can be completed is defined to mean that there exists a mapping h: W -+ U making this diagram commutative.

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 . Proal. I: Pow A Deny the statement. Then there exists an injection A. On the other hand, there is an injection ~ g {A x ~PowA ~ {x}. From the (Cantor-)SchrOder-Bemstein theorem, deduce that Pow A ~ A, which contradicts Cantor's theorem. 0 The statement shows that there is no such thing as a "largest" set; that is, there is no set into which every other set can be injected.

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Algebra: Rings, Modules and Categories I by Carl Faith


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