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Math 6250: Theory of Rings, Spring 2012


Syllabus


Course description: 
This course will be an introduction to the theory of noncommutative rings. I expect to cover semisimple rings, the Jacobson radical, group rings, representations of finite groups, the density theorem, the double centralizer theorem, central simple algebras, division rings, and the Brauer group.

Prerequisites:
MATH 6130, 6140.

Text: 
A First Course in Noncommutative Rings, 2nd ed, by T. Y. Lam (2001).

Other Fine Books on the Subject: 
  ♦ Lectures on Modules and Rings (Grad. Texts in Math. 189), T. Y. Lam (1998).
  ♦ Ring Theory, Vol. I, Louis Rowen (1988).
  ♦ Associative algebras (Grad. Texts in Math. 88) Richard Pierce (1982).
  ♦ Noncommutative Rings (Carus Monographs 15), I. N. Herstein (1968).
  ♦ Structure of Rings, 2nd ed (AMS Colloquium Publications 37), Nathan Jacobson (1964).

Books on Related Topics: 
  ♦ Abstract Algebra, 3rd ed, (Chapters 7, 18 and 19), by Dummit and Foote.
  ♦ Categories for the Working Mathematician, by MacLane (1971). (2nd ed. (1998).)

Homework: 
If you are enrolled for a grade I will ask you to solve some problems. You will be asked to work on the problems in small groups of 2-3. Different groups will be assigned different problems. You will typically have a week for your group to solve its assigned problems and submit the solutions. (This deadline is not strict. but I'll check in with you if I don't get solutions within a couple days of the due date.) All members of a group should review each solution from the group before it is submitted, and should agree that the solution is correct and well written.

If you are solving Problem M of HW N, please submit the solution as a PDF file called ringsNpM.pdf. Your writeup should include the assignment number, the problem number, the problem statement, and the names of the group members. Each problem solution should be submitted as its own document, not one document containing several problem solutions. After receiving your solution I will act as the referee on your submission. This means that I will check it for clarity and correctness, and I will correspond with you about improvements. This step in the process should take at most one week. You are not obligated to take any of my advice, but if one of my comments involves a correction, then you should correct that part in some way. (For example, if I say "Here is a shorter way to do it", you don't have to change anything unless you want to. If I say "The first displayed equation is wrong. Here is how to fix it …", then you should fix the error, not necessarily along the lines of my suggestion.)

New assignments will be posted every other week starting the second week. Solutions to old assignments will be posted when they are in final form. You should read your classmates' solutions.