A commutative analogue of the group ring

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