A commutative analogue of the group ring
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.
scientific article; zbMATH DE number 5146869
scientific article