Representations over \(G\)-rings and cohomology (Q1078671)

Let \(G\) be a group. A \(G\)-ring is a ring \(\Lambda\) together with a \(G\)-action on \(\Lambda\) preserving the ring structure. Then we introduce a Grothendieck group R(G,\(\Lambda)\) associated with the abelian semi-group consisting of representations over \(\Lambda\). The group \(R(G,\Lambda)\) is a generalization of the representation rings \(R(G)\) and \(RO(G)\). The purpose of the present paper is to express \(R(G,\Lambda)\) in terms of the cohomology \(H^ 1(G;GL(n,\Lambda))\) of the group G with coefficients in a non-abelian group \(GL(n,\Lambda)\) in the sense of Serre.