Author | Lam, T. Y. author |
---|---|

Title | A First Course in Noncommutative Rings [electronic resource] / by T. Y. Lam |

Imprint | New York, NY : Springer US, 1991 |

Connect to | http://dx.doi.org/10.1007/978-1-4684-0406-7 |

Descript | XV, 397p. online resource |

SUMMARY

One of my favorite graduate courses at Berkeley is Math 251, a one-semester course in ring theory offered to second-year level graduate students. I taught this course in the Fall of 1983, and more recently in the Spring of 1990, both times focusing on the theory of noncommutative rings. This book is an outgrowth of my lectures in these two courses, and is intended for use by instructors and graduate students in a similar one-semester course in basic ring theory. Ring theory is a subject of central importance in algebra. Historically, some of the major discoveries in ring theory have helped shape the course of development of modern abstract algebra. Today, ring theory is a ferยญ tile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential opยญ erators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups). In view of these basic connections between ring theory and other branches of mathematยญ ics, it is perhaps no exaggeration to say that a course in ring theory is an indispensable part of the education for any fledgling algebraist. The purpose of my lectures was to give a general introduction to the theory of rings, building on what the students have learned from a stanยญ dard first-year graduate course in abstract algebra

CONTENT

1. Wedderburn-Artin Theory -- ยง1. Basic terminology and examples -- ยง2. Semisimplicity -- ยง3. Structure of semisimple rings -- 2. Jacobson Radical Theory -- ยง4. The Jacobson radical -- ยง5. Jacobson radical under change of rings -- ยง6. Group rings and the J-semisimplicity problem -- 3. Introduction to Representation Theory -- ยง7. Modules over finite-dimensional algebras -- ยง8. Representations of groups -- ยง9. Linear groups -- 4. Prime and Primitive Rings -- ยง10. The prime radical; prime and semiprime rings -- ยง11. Structure of primitive rings; the Density Theorem -- ยง12. Subdirect products and commutativity theorems -- 5. Introduction to Division Rings -- ยง13. Division rings -- ยง14. Some classical constructions -- ยง15. Tensor products and maximal subfields -- ยง16. Polynomials over division rings -- 6. Ordered Structures in Rings -- ยง17. Orderings and preorderings in rings -- ยง18. Ordered division rings -- 7. Local Rings, Semilocal Rings, and Idempotents -- ยง19. Local rings -- ยง20. Semilocal rings -- ยง21. The theory of idempotents -- ยง22. Central idempotents and block decompositions -- 8. Perfect and Semiperfect Rings -- ยง23. Perfect and semiperfect rings -- ยง24. Homological characterizations of perfect and semiperfect rings -- ยง25. Principal indecomposables and basic rings -- References -- Name Index

Mathematics
Algebra
Mathematics
Algebra