A generalization of the smash product of a graded ring

Abstract

For any group G and G-graded ring R, there exists a ring $\text{S = R ♯ G}{}^{\mathrm{\ast }}$, defined analogously to the smash product of R with the dual of the group ring for finite G, such that the categories of unital S-modules and G-graded R-modules are isomorphic. The category of unital S-modules is equivalent to the category of A-modules for a ring A with identify if and only if S is finitely generated as an S-S bimodule. Finally the category isomorphism is applied to obtain a characterization of the graded Jacobson radical for infinite G.