This new book can be read independently from the first volume and may be used for lecturing, seminar- and self-study, or for general reference. It focuses more on specific topics in order to introduce readers to a wealth of basic and useful ideas without the hindrance of heavy machinery or undue abstractions. User-friendly with its abundance of examples illustrating the theory at virtually every step, the volume contains a large number of carefully chosen exercises to provide newcomers with practice, while offering a rich additional source of information to experts. A direct approach is used in order to present the material in an efficient and economic way, thereby introducing readers to a considerable amount of interesting ring theory without being dragged through endless preparatory material.
Rezensionen / Stimmen
"As in the previous book, the presentation is extremely clear and incisive and the exercises are both interesting and challenging. Without a doubt, the book could indeed serve as a textbook for a very good---though definitely formidable---graduate course in ring theory. The author is the first to admit, in his introduction, that he is far from having covered all of the interesting topics in noncommutative ring theory. Therefore, given the high level of presentation of the two books he has already published, one can only hope that a third volume will be in the offing, despite his "solemn pledge" to the contrary."--MATHEMATICAL REVIEWS
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Höhe: 235 mm
Breite: 155 mm
Dicke: 37 mm
Gewicht
ISBN-13
978-0-387-98428-5 (9780387984285)
DOI
10.1007/978-1-4612-0525-8
Schweitzer Klassifikation
1 Free Modules, Projective, and Injective Modules.- 1. Free Modules.- 1A. Invariant Basis Number (IBN).- 1B. Stable Finiteness.- 1C. The Rank Condition.- 1D. The Strong Rank Condition.- 1E. Synopsis.- Exercises for &x00A7;1.- 2. Projective Modules.- 2A. Basic Definitions and Examples.- 2B. Dual Basis Lemma and Invertible Modules.- 2C. Invertible Fractional Ideals.- 2D. The Picard Group of a Commutative Ring.- 2E. Hereditary and Semihereditary Rings.- 2F. Chase Small Examples.- 2G. Hereditary Artinian Rings.- 2H. Trace Ideals.- Exercises for &x00A7;2.- 3. Injective Modules.- 3A. Baer's Test for Injectivity.- 3B. Self-Injective Rings.- 3C. Injectivity versus Divisibility.- 3D. Essential Extensions and Injective Hulls.- 3E. Injectives over Right Noetherian Rings.- 3F. Indecomposable Injectives and Uniform Modules.- 3G. Injectives over Some Artinian Rings.- 3H. Simple Injectives.- 31. Matlis' Theory.- 3J. Some Computations of Injective Hulls.- 3K. Applications to Chain Conditions.- Exercises for &x00A7;3.- 2 Flat Modules and Homological Dimensions.- 4. Flat and Faithfully Flat Modules.- 4A. Basic Properties and Flatness Tests.- 4B. Flatness, Torsion-Freeness, and von Neumann Regularity.- 4C. More Flatness Tests.- 4D. Finitely Presented (f.p.) Modules.- 4E. Finitely Generated Flat Modules.- 4F. Direct Products of Flat Modules.- 4G. Coherent Modules and Coherent Rings.- 4H. Semihereditary Rings Revisited.- 41. Faithfully Flat Modules.- 4J. Pure Exact Sequences.- Exercises for &x00A7;4.- 5. Homological Dimensions.- 5A. Schanuel's Lemma and Projective Dimensions.- 5B. Change of Rings.- 5C. Injective Dimensions.- 5D. Weak Dimensions of Rings.- 5E. Global Dimensions of Semiprimary Rings.- 5F. Global Dimensions of Local Rings.- 5G. Global Dimensions of Commutative Noetherian Rings.- Exercises for &x00A7;5.- 3 More Theory of Modules.- 6. Uniform Dimensions, Complements, and CS Modules.- 6A. Basic Definitions and Properties.- 6B. Complements and Closed Submodules.- 6C. Exact Sequences and Essential Closures.- 6D. CS Modules: Two Applications.- 6E. Finiteness Conditions on Rings.- 6F. Change of Rings.- 6G. Quasi-Injective Modules.- Exercises for &x00A7;6.- 7. Singular Submodules and Nonsingular Rings.- 7A. Basic Definitions and Examples.- 7B. Nilpotency of the Right Singular Ideal.- 7C. Goldie Closures and the Reduced Rank.- 7D. Baer Rings and Rickart Rings.- 7E. Applications to Hereditary and Semihereditary Rings.- Exercises for &x00A7;7.- 8. Dense Submodules and Rational Hulls.- 8A. Basic Definitions and Examples.- 8B. Rational Hull of a Module.- 8C. Right Kasch Rings.- Exercises for &x00A7;8.- 4 Rings of Quotients.- 9. Noncommutative Localization.- 9A. "The Good'.- 9B. "The Bad'.- 9C. "The Ugly".- 9D. An Embedding Theorem of A. Robinson.- Exercises for &x00A7;9.- 10. Classical Rings of Quotients.- 10A. Ore Localizations.- 10B. Right Ore Rings and Domains.- 10C. Polynomial Rings and Power Series Rings.- 10D. Extensions and Contractions.- Exercises for §10.- 11. Right Goldie Rings and Goldie's Theorems.- 11A. Examples of Right Orders.- 11B. Right Orders in Semisimple Rings.- 11C. Some Applications of Goldie's Theorems.- 11D. Semiprime Rings.- 11E. Nil Multiplicatively Closed Sets.- Exercises for &x00A7;11.- 12. Artinian Rings of Quotients.- 12A. Goldie's ?-Rank.- 12B. Right Orders in Right Artinian Rings.- 12C. The Commutative Case.- 12D. Noetherian Rings Need Not Be Ore.- Exercises for &x00A7;12.- 5 More Rings of Quotients.- 13. Maximal Rings of Quotients.- 13A. Endomorphism Ring of a Quasi-Injective Module.- 13B. Construction of Qrmax(R).- 13C. Another Description of Qrmax(R).- 13D. Theorems of Johnson and Gabriel.- Exercises for §13.- 14. Martindale Rings of Quotients.- 14A. Semiprime Rings Revisited.- 14B. The Rings Qr(R) and Qs(R).- 14C. The Extended Centroid.- 14D. Characterizations of and Qr(R) and Qs(R).- 14E. X-Inner Automorphisms.- 14F. A Matrix Ring Example.- Exercises for &x00A7;14.- 6 Frobenius and Quasi-Frobenius Rings.- 15. Quasi-Frobenius Rings.- 15A. Basic Definitions of QF Rings.- 15B. Projectives and Injectives.- 15C. Duality Properties.- 15D. Commutative QF Rings, and Examples.- Exercises for §15.- 16. Frobenius Rings and Symmetric Algebras.- 16A. The Nakayama Permutation.- 16B. Definition of a Frobenius Ring.- 16C. Frobenius Algebras and QF Algebras.- 16D. Dimension Characterizations of Frobenius Algebras.- 16E. The Nakayama Automorphism.- 16F. Symmetric Algebras.- 16G. Why Frobenius?.- Exercises for &x00A7;16.- 7 Matrix Rings, Categories of Modules, and Morita Theory.- 17. Matrix Rings.- 17A. Characterizations and Examples.- 17B. First Instance of Module Category Equivalences.- 17C. Uniqueness of the Coefficient Ring.- Exercises for &x00A7;17.- 18. Morita Theory of Category Equivalences.- 18A. Categorical Properties.- 18B. Generators and Progenerators.- 18C. The Morita Context.- 18D. Morita I, II, III.- 18E. Consequences of the Morita Theorems.- 18F. The Category ? [M].- Exercises for &x00A7;18.- 19. Morita Duality Theory.- 19A. Finite Cogeneration and Cogenerators.- 19B. Cogenerator Rings.- 19C. Classical Examples of Dualities.- 19D. Morita Dualities: Morita I.- 19E. Consequences of Morita I.- 19F. Linear Compactness and Reflexivity.- 19G. Morita Dualities: Morita II.- Exercises for &x00A7;19.- References.- Name Index.
