Project Report

I researched group rings during my stay in Korea. A group ring is basically a vectorspace with a basis that forms a group under multiplication. A group ring, denoted RG, has a base ring, R, and G denotes its group of basis elements. One main question in the study of group rings is: If R has a certain property, say property P, what conditions on the group G are necessary and/or su?cient in order for RG to have property P? For example, a classical theorem for group rings is the following: RG is Artinian if and only if R is Artinian and G is ?nite. There are theorems and surprisingly many open problems of this ?avor. I studied group rings because the theory is very useful for ?nding examples and counterexamples to conjectures. In particular, I was interested in algebraic properties related to Baer rings. In such rings, there are a lot of idempotents, which are crucial ingredients for decomposing rings into simpler pieces. The general idea is we want to take a possibly complicated object and break it down to smaller, better understood pieces. A celebrated example of this idea is the following: Every semisimple Artinian ring is a ?nite direct sum of matrix rings over division rings. Matrix rings over division rings are very well understood objects in linear algebra, so what the previous example does is reduce a possibly quite complicated thing into a ?nite number of simple things. I also attended the sixth China-Japan-Korea International Conference on Ring Theory in Suwon.There I met many mathematicians, mostly from East Asia, but also from India, Iran, Europe, and some other Americans attended as well. I heard a lot of interesting talks, and received some very helpful advice for my research from Prof. Park and others at the conference. I also gave a talk at the conference on a paper that my Ph.D. advisor and I hope to submit to a journal soon.

Agency
National Science Foundation (NSF)
Institute
Office of International and Integrative Activities (IIA)
Application #
1106903
Program Officer
Carter Kimsey
Project Start
Project End
Budget Start
2011-06-01
Budget End
2012-05-31
Support Year
Fiscal Year
2011
Total Cost
$5,700
Indirect Cost
Name
Ryan Christopher E
Department
Type
DUNS #
City
Lafayette
State
LA
Country
United States
Zip Code
70504