A commutative analogue of the group ring

Woodcock, Chris F. (2007) A commutative analogue of the group ring. Journal of Pure and Applied Algebra, 210 (1). pp. 193-199. ISSN 0022-4049. (The full text of this publication is not available from this repository)

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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.

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Pure Mathematics
Depositing User: Chris F Woodcock
Date Deposited: 04 Jun 2008 16:39
Last Modified: 29 Apr 2014 14:52
Resource URI: http://kar.kent.ac.uk/id/eprint/3170 (The current URI for this page, for reference purposes)
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