Semiperfect and Perfect Group Rings

Abstract

In this thesis, we give a survey of necessary and sufficient conditions on a group G and a ring R for the group ring RG to be semiperfect and perfect. A ring R is called semiperfect R/RadR is semisimple and idempotents of R/RadR can be lifted to R. It is given that if RG is semiperfect, so is R. Necessary conditions on G for RG to be semiperfect are also given for some special type of groups. For the sufficient conditions, several types of rings and groups are considered. If R is commutative and G is abelian, a complete characterization is given in terms of the polynomial ring R[X]. A ring R is called left (respectively, right) perfect if R/Rad R is semisimple and Rad R is left (respectively, right) T-nilpotent. Equivalently, a ring is called left (respectively, right) perfect if R satisfies the descending chain condition on principal right (respectively, left) ideals. By using these equivalent definitions of a perfect ring and results from group theory, a complete characterization of a perfect group ring RG is given in terms of R and G.