COURSE SYLLABUS

Course Number: MATH 7340 
Course Title: Ring Theory
Credit Hours: 3
Prerequisites: MATH 7320 
Corequisite: 

Objectives: An overview of the theory of Commutative Rings or Noncommutative Rings (in particular Algebras). Whether the Commutative or Noncommutative theory is emphasized is left to the discretion of the instructor. 

Course Content:
Commutative Rings.
Multiplicative subsets, localization and ideals in localizations [2 days]
Integral extensions, integral closure, prime ideals in integral extensions, Cohen-Seidenberg Theorems (lying over, going up and going down) [4 days]
Chain conditions, the structure of Artinian rings, Nakayama's Lemma, Krull Intersection Theorem [4 days]
Invertible ideals, Dedekind domains, integral extensions of Dedekind domains [5 days]
Primary decomposition, zero divisors, heights and depths of ideals, Krull Principal ideal Theorem and its generalization [7 days]
Hilbert Rings, Noether Normalization and Hilbert's Nullstellensatz [7 days]
Applications to Algebraic Geometry: varieties, coordinate rings, morphisms, birational equivalence, regular and singular points, resolution of singularities, and dimension. [5 days]
R-sequences, grade and Macaulay rings [8 days]

Noncommutative Rings and Algebras.
Examples: group algebras, endomorphism algebras, matrix algebras, finite dimensional algebras over a field, quaternion algebras [4 days]
Semisimple Modules, chain conditions, Wedderburn's Structure Theorem (including uniqueness), Maschke's Theorem [5 days]
Projective modules over Artinian algebras [7 days]
Finite representation type and Roiter's Theorem [7 days]
Separable algebras [5 days]
Central simple algebras, the Density Theorem, Brauer groups and the Noether-Skolem Theorem [5 days]
Cyclic Division Algebras and Division Algebras over local fields [8 days] 

Possible textbooks:
Commutative Rings, I. Kaplansky, 1974, University of Chicago Press, Chicago and London
Commutative Ring Theory, H. Matsumura, 1986, Cambridge University Press, Cambridge and New York
Associative Algebras, R. S. Pierce, 1982, Springer-Verlag, New York

Sample Grading and Evaluation Procedures

A graduate student is expected to creatively engage the mathematical material of the course.  The student will be given problems to solve; these problems may include the derivation of proofs to theorems.  These solutions may be presented in class on the blackboard or be written up to be handed in to the instructor. Extended projects may also be assigned.  

Grade Calculation
Presentation of solutions to problems/theorems, homework:		40%
Midterm exam or midterm project:				25%
Final Exam or culminating project:				35%

There may be variations in these procedures depending on the individual instructors and the nature of the specific material.
Sample Statement Re: Accommodations
	Students who need accommodations are asked to arrange a meeting during office hours the first week of classes, or as soon as possible if accommodations are needed immediately.  If you have a conflict with my office hours, an alternate time can, be arranged.  To set up this meeting, please contact me by E-mail.  Bring a Copy of your Accommodation Memo and an Instructor Verification Form to the meeting.  If you do not have an Accommodation Memo but need accommodations, make an appointment with The Program for Students with Disabilities, 1244 Haley Center, 844-2096 (V/TT).  (Note: Instructor office room, office hours and email address will be made available on the course syllabus and on the first day of class.)

JUSTIFICATION FOR GRADUATE CREDIT

	This course is part of a modified semester version of a 600-level course under the quarter system.  Under the quarter system it was specifically designed as a graduate course.  It was approved as part of our graduate program and has been a traditional part of our graduate program offerings.  Outside of modernization, the standard of the course remains at the same graduate level that the department has maintained in the past. The course demands considerable mathematics background and a degree of mathematical maturity traditionally found at the graduate level.  The 7000-level course will inculcate the same analytical skills and depth of understanding previously demanded by the comparable 600-level quarter course.  In order to successfully complete the course the student will have to demonstrate an ability to creatively examine and apply the mathematics presented in the course.