A commutative analogue of the group ring

Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded image-algebra RG. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. In the case when G is an elementary Abelian p-group it turns out that RG is closely related to the symmetric algebra over image of the dual of G. We intend in subsequent papers to explore the close relationship between G and RG in the case of a general (possibly non-Abelian) group G. Here we show that the Krull dimension of RG is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1.