I have previously experienced cohomology group and cohomology ring.
Today someone suggested me to look over the concepts of cohomology field. I think what is meant there is that the cohomology $H^n(G,C)$, the coefficient $C$ can be a group, a ring or generalized to a field.
My question is that :
what are the advantages to look at Cohomology group vs Cohomology ring vs Cohomology field?
Is it possible (always possible or when is it possible) to generalize a Cohomology group to a Cohomology ring, to a Cohomology field?
see also What are cohomology rings good for?