Algebra

Rings, Modules and Categories I

Carl Faith (Autor)

Buch | Softcover

XXIV, 568 Seiten

2012 | 1. Softcover reprint of the original 1st ed. 1973
Springer Berlin (Verlag)
978-3-642-80636-0 (ISBN)

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VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction.

to Volume.- Foreword on Set Theory.- I Introduction to the Operations: Monoid, Semigroup, Group, Category, Ring, and Module.- 1. Operations: Monoid, Semigroup, Group, and Category.- 2. Product and Coproduct.- 3. Ring and Module.- 4. Correspondence Theorems for Projective Modules and the Structure of Simple Noetherian Rings.- 5. Limits, Adjoints, and Algebras.- 6. Abelian Categories.- II Structure of Noetherian Semiprime Rings.- 7. General Wedderburn Theorems.- 8. Semisimple Modules and Homological Dimension.- 9. Noetherian Semiprime Rings.- 10. Orders in Semilocal Matrix Rings.- III Tensor Algebra.- 11. Tensor Products and Flat Modules.- 12. Morita Theorems and the Picard Group.- 13. Algebras over Fields.- IV Structure of Abelian Categories.- 14. Grothendieck Categories.- 15. Quotient Categories and Localizing Functors.- 16. Torsion Theories, Radicals, and Idempotent, Topologizing, and Multiplicative Sets.

Erscheint lt. Verlag	3.8.2012
Reihe/Serie	Grundlehren der mathematischen Wissenschaften
Zusatzinfo	XXIV, 568 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	823 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Algebra • Autodesk Maya • Character • Class • Commutative Ring • Coproduct • Group • Kategorie • Kategorie (Math.) • Matrix • Modul • Ring • Ring (Math.) • Ring Theory • Semigroup • Theorem • Torsion
ISBN-10	3-642-80636-8 / 3642806368
ISBN-13	978-3-642-80636-0 / 9783642806360
Zustand	Neuware