A commutative analogue of the group ring

Abstract

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 .