Algebra: Rings, Modules and Categories I.

FAITH, Carl

ISBN 10: 3540055517 / ISBN 13: 9783540055518

Editore: Springer, Berlin, 1973

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Bln., Springer 1973. gr.8°. XXII, 565 p. OCloth. with dust jacket. (edge sligtly foxed).- Grundlehren der mathematischen Wissenschaften, 190.- Incl. bibliography.- Name on flyleaf.

Codice articolo 9979

Dati bibliografici

• Titolo: Algebra: Rings, Modules and Categories I.
• Editore: Springer, Berlin
• Data di pubblicazione: 1973
• Lingua: Inglese
• ISBN 10: 3540055517
• ISBN 13: 9783540055518
• Rilegatura: Hardcover
• Condizione: Gut

Descrizione articolo

Riassunto

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.

Contenuti

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.