{"product_id":"algebra-rings-modules-and-categories-i-9783642806360","title":"Algebra: Rings, Modules and Categories I","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eVI 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003eto 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.","brand":"Springer-Verlag Berlin and Heidelberg GmbH \u0026 Co. KG","offers":[{"title":"Default Title","offer_id":51360443466071,"sku":"9783642806360","price":75.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9783642806360.jpg?v=1754127628","url":"https:\/\/bookcurl.com\/products\/algebra-rings-modules-and-categories-i-9783642806360","provider":"Book Curl","version":"1.0","type":"link"}