A commutative analogue of the group ring
type: Item
instance of: scholarly article
title: A commutative analogue of the group ring
zbMATH Open document ID: 1121.13011
author: Chris Woodcock
published in: Journal of Pure and Applied Algebra
publication date: 2007-04-26T00:00:00Z
review text: Let \(R\) be a commutative ring with identity and let \(G\) be a finite group. The author defines a \(\otimes\) multiplication on the free \(R\)-module \(R[G]\) of rank \(| G| \) to be a binary operation satisfying (1) \(R[G], \otimes\) is a commutative \(R\)-algebra with identity. (2) \(\otimes\) is a 2-cocycle with respect to the usual direct sum multiplication \(*\) on \(R[G]\), that is \((a*b)\otimes c+(a\otimes b)*c=a*(b\otimes c)+a\otimes (b*c)\) for all \(a,b,c\in R[G]\). (3) \(\otimes\) is \(G\)-invariant, that is \(x_g \cdot (a\otimes b)=(x_g \cdot a)\otimes (x_g \cdot b)\) for all \(a,b\in R[G]\) and \(g\in G\). Thus a \(\otimes \) multiplication is a \(G\)-invariant commutative \(R\)-algebra multiplication on the group algebra \(R[G]\) which is a cocycle with respect to the standard multiplication on \(R[G]\) and has the same identity element. Woodcock then defines a commutative graded \(\mathbb Z\)-algebra \(R_G\) and shows (Theorem 3.3) that there is a natural bijection from the set of all ring homomorphisms \(\phi :R_G \to R\) onto the set of all \(\otimes\)-multiplications on \(R[G]\). The author goes on to demonstrate (Theorem 5.3) 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'' (from the prime field) in which case the Krull dimension is \(r+1\). In the case when \(G\) is an elementary abelian \(p\)-group, \(R_G\) is closely related to the symmetric algebra over \({\mathbb F}_p\) of the dual of \(G\). Woodcock plans to explore the relation between \(G\) and \(R_G\) for more general groups \(G\) in future papers.
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13D10
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5146869
zbMATH Keywords: group ring
zbMATH Keywords: graded algebra
zbMATH Keywords: \(G\)-invariant, deformation
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.09.011
OpenAlex ID: W2041881861
cites work: Q3708174
cites work: Q5563439
cites work: On the deformation of rings and algebras
cites work: Q5569508
cites work: The spectrum of an equivariant cohomology ring. I. II
cites work: The Automorphism Group of a p -Adic Convolution Algebra
DOI: 10.1016/J.JPAA.2006.09.011
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description: scientific article; zbMATH DE number 5146869