Coprime packedness and set theoretic complete intersections of ideals in polynomial rings

Abstract

A ring is called coprimely packed if every ideal contained in a union of maximal ideals is already contained in one of them. For a ring \(R\) let \(R\langle x\rangle\) denote the localization of \(R[x]\) at the set of monic polynomials. The author shows for any Noetherian normal domain \(R\) that \(R\langle x\rangle\) is coprimely packed if and only if \(R\) is coprimely packed and of dimension at most one. Furthermore, again for a Noetherian normal domain \(R\), he shows that the following conditions are equivalent: (i) \(R\langle x\rangle\) is coprimely packed; (ii) \(R\langle x\rangle\) is a Dedekind domain with torsion ideal class group; (iii) \(R\) is a Dedekind domain with torsion ideal class group; (iv) \(R\) is of dimension one and each proper prime ideal of \(R[x]\) is a set-theoretic complete intersection. Similar criteria are proved for \(R\) a Noetherian arithmetical ring and also for \(R\) a one-dimensional Bézout domain.