Textbook writing must be one of the cruelest of self-inflicted tortures. - Carl Faith Math Reviews 54: 5281 So why didn't I heed the warning of a wise colleague, especially one who is a great expert in the subject of modules and rings? The answer is simple: I did not learn about it until it was too late! My writing project in ring theory started in 1983 after I taught a year-long course in the subject at Berkeley. My original plan was to write up my lectures and publish them as a graduate text in a couple of years. My hopes of carrying out this plan on schedule were, however, quickly dashed as I began to realize how much material was at hand and how little time I had at my disposal. As the years went by, I added further material to my notes, and used them to teach different versions of the course. Eventually, I came to the realization that writing a single volume would not fully accomplish my original goal of giving a comprehensive treatment of basic ring theory. At the suggestion of Ulrike Schmickler-Hirzebruch, then Mathematics Editor of Springer-Verlag, I completed the first part of my project and published the write up in 1991 as A First Course in Noncommutative Rings, GTM 131, hereafter referred to as First Course (or simply FC).
Inhaltsverzeichnis
1 Free Modules, Projective, and Injective Modules. - 1. Free Modules. - 2. Projective Modules. - 3. Injective Modules. - 31. Matlis Theory. - 2 Flat Modules and Homological Dimensions. - 4. Flat and Faithfully Flat Modules. - 41. Faithfully Flat Modules. - 5. Homological Dimensions. - 3 More Theory of Modules. - 6. Uniform Dimensions, Complements, and CS Modules. - 7. Singular Submodules and Nonsingular Rings. - 8. Dense Submodules and Rational Hulls. - 4 Rings of Quotients. - 9. Noncommutative Localization. - 10. Classical Rings of Quotients. - 11. Right Goldie Rings and Goldie s Theorems. - 12. Artinian Rings of Quotients. - 5 More Rings of Quotients. - 13. Maximal Rings of Quotients. - 14. Martindale Rings of Quotients. - 6 Frobenius and Quasi-Frobenius Rings. - 15. Quasi-Frobenius Rings. - 16. Frobenius Rings and Symmetric Algebras. - 7 Matrix Rings, Categories of Modules, and Morita Theory. - 17. Matrix Rings. - 18. Morita Theory of Category Equivalences. - 19. Morita Duality Theory. - References. - Name Index.
