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 Z-algebra R-G. 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 R-G is closely related to the symmetric algebra over F-p of the dual of G. We intend in subsequent papers to explore the close relationship between G and R-G in the case of a general (possibly non-Abelian) group G.

Here we show that the Krull dimension of R-G 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.