Quotients of unit groups of commutative rings (Q2785909)

Given a commutative ring with identity, denote by \(U(R)\) the group of units of its total quotient ring. Let \(R\subset S\) be a ring extension. The authors deal with the following two questions: NEWLINENEWLINENEWLINE(1) When is the quotient \(U(S)/U(R)\) of the unit groups finitely generated? NEWLINENEWLINENEWLINE(2) When does \(U(S)/U(R)\) finite or finitely generated imply that \(S\) is a finitely generated \(R\)-module? NEWLINENEWLINENEWLINEWith several results that provide some answers to these questions there is also the following conjecture: NEWLINENEWLINENEWLINELet \(K\) be an infinite field and \(S\) a finitely generated \(K\)-algebra. Then \(U(S)/K^*\) is finitely generated if and only if \(K\) is algebraically closed in \(S\) (\(K^*\) is the multiplicative group of \(K\)).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].