Group rings which are Jacobson rings
Let A be a commutative ring and G an abelian group. In [\textit{G. Karpilovsky}, Commutative group algebras (1983; Zbl 0508.16010)] the following problem was stated (Problem 1): Find necessary and sufficient conditions for the group ring AG to be a Jacobson ring. In the case where the torsion-free rank of G is finite it is easy to prove that AG is a Jacobson ring if and only if A is. For a domain R let us denote by \(\hat R\) the field of fractions of R, and by fr(R) the smallest cardinality of a set of generators of \(\hat R\) as R-algebra. Our main result is Theorem: Let G be a group of infinite torsion-free rank \(\alpha\). Then AG is a Jacobson ring if and only if the following conditions are satisfied: 1) all homomorphic images of A have cardinality exceeding \(\alpha\), 2) for any prime homomorphic image B of A such that \(fr(B)\leq \alpha\) and for any ring C, \(B\subset C\subset \hat B,\) we have \(fr(C)\neq 1.\) The proof goes through a reduction to polynomial rings and is based on the fact that fr(R) is inherited by integral extensions of R.
scientific article; zbMATH DE number 3849427