We are concerned in this paper
with the following question: When is the maximal right quotient ring of the group
algebra kG a right self-injective ring? In general, the maximal right quotient ring
Q(R) of a ring R is a right R-submodule of the right injective hull E(R) of
R, and we may rephrase our question as: When does Q(kG) = E(kG)? Of
course, a sufficient condition for this to occur is that kG be right nonsingular,
so that, for example, E(kG) = Q(kG) when k is a field of characteristic
zero. However, Q(kG) is often injective even when kG is a singular ring; for
example, when G is finite, it is well-known that kG is itself an injective
ring.
