Coefficient rings of multidimensional torus extensions
All rings below are assumed to be commutative. Let G be a torsion-free abelian group. A ring R is said to be G-invariant if for any ring S, \(R\cong S\) whenever the group rings RG and SG are isomorphic. A ring R is strongly G-invariant provided for any ring S and any isomorphism \(\psi\) : RG\(\to SG\) there exists an S-automorphism \(\sigma\) : SG\(\to SG\) such that \(\sigma\psi\) is a G-homomorphism (i.e. \(\sigma \psi (g)=g\) for all \(g\in G)\). Finally, R is called totally G-invariant if for any ring S and any isomorphism \(\psi\) : RG\(\to SG\), we have \(\psi (R)=S.\) The main purpose of this paper is to prove that all results concerning the three types of invariance which are valid for the infinite cyclic group remain true for an arbitrary free abelian group of finite rank. The paper contains a number of interesting result of which we quote the following Theorem: Let G be a free abelian group of finite rank. A ring R is strongly G-invariant if and only if for any ring S and any isomorphism \(\psi\) : RG\(\to SG\), we have \(\psi\) (U(R))\(\subseteq U(S)+N(SG)\) where N(SG) is the nilradical of SG. Corollary. A ring R is totally G-invariant if and only if for any ring S and any isomorphism \(\psi\) : RG\(\to SG\), we have \(\psi\) (R)\(\subseteq S\).
scientific article; zbMATH DE number 3975182
scientific article