Finite Dimensional Group Rings

A Ring is Right Finite Dimensional If It Contains No Infinite Direct Sum of Right Ideals. We Prove that If a Group G is Finite, Free Abelian, or Finitely Generated Abelian, then a Ring R is Right Finite Dimensional If and Only If the Group Ring RG is Right Finite Dimensional. a Ring R is a Self-Injective Cogenerator Ring If Rn is Injective and RR is a Cogenerator in the Category of Unital Right /{-Modules; This Means that Each Right Unital A-Module Can Be Embedded in a Direct Product of Copies of R. Let G Be a Finite Group Where the Order of G is a Unit in R. Then the Group Ring RG is a Selfinjective Cogenerator Ring If and Only If R is a Self-Injective Cogenerator Ring. Additional Applications Are Given. © 1973 American Mathematical Society.