Title: |
Mathematics 724-725: Commutative Algebra, Non-Commutative Algebra |
Credits: |
3 credits each |
Prerequesites: |
Mathematics 721 or Departmental approval |
Required for: |
|
Suggested Texts: |
Associative Algebras by R. S. Pierce Ring Theory, Volume I and II by L. H. Rowen A first course in non commutative rings by T. Y. Lam |
Supplemental Texts: |
Identities by Yu. A. Bakhturin and A. Yu. Ol'shanskij Noncommutative rings by L. A. Bokhut', I. V. L'vov, and V. K. Kharchenko Representation theory of finite groups and associative algebras by C. Curtis and I. Reiner Methods in representation theory by C. Curtis and I. Reiner Algebra II: Ring Theory by C. Faith Noncommutative rings by I. Herstein Basic Algebra II by N. Jacobson Rings with polynomial identities by C. Procesi |
Commutative Algebra: A topics course in commutative algebra. Selected topics from the ideal theory of commutative rings, localizations and local methods, Noetherian rings, polynomial and power series rings, valuation theory, integrality and its generalizations, factorization, rings of special type, homological and topological aspects of ring theory.
Non-Commutative Algebra:
| Artin-Wedderburn Theorem |
| Morita Theory |
| Central simple algebras |
| Representation theory |
| Polynomial Identities |
| Jacobson Density Theorem |
| Artin-Wedderburn Theorem |
| Morita theorem |
| Jacobson-Bourbaki theorem |
| Azumaya algebras |
| Brauer group |
| Maschke theorem |
| Characters |
| Frobenius reciprocity theorem |
| Quivers |
| Artin-Procesi theorem |
| The algebra of generic matrices |
This course constitutes an introduction to the theory of noncommutative rings.