Categories of modules graded by G-sets. Applications

AbstractThis paper studies rings R graded by a G-set X such that the category of X-graded left R-modules, (G,X,R)-gr, is equivalent to A-mod, A a ring with 1. Examples of such rings, other than strongly graded rings, exist. For every infinite group G, there is a G-graded ring R such that (G,G,R)-gr = R-gr is equivalent to A-mod, for some A with 1, but R contains no strongly graded subring except Re. The final section gives connections between some properties of R and Re for such rings