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

摘要

A ring R is said to be coprimely packed if whenever I is an ideal of R and S is a set of maximal ideals of R with I subset of or equal to boolean OR{M is an element of S}, then I subset of or equal to M for some M is an element of S. Let R be a ring and R[X] be the localization of R[X] at its set of monic polynomials. We prove that if R is a Noetherian normal domain, then the ring R[X] is coprimely packed if and only if R is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of R[X] is a set theoretic complete intersection. A similar result is also proved when R is either a Noetherian arithmetical ring or a Bezout domain of dimension one.