Coprimely packed rings

Abstract

An ideal I of a commutative ring R with identity is said to be coprimely packed by prime ideals of R if whenever I is coprime to each element of a family of prime ideals of R, I is not contained in the union of prime ideals of the family. We say that R is coprimely packed if every ideal of R is coprimely packed. It is shown that in a Noetherian arithmetical ring R every prime ideal is coprimely packed if and only if a positive power of every ideal of R is principal. Consequently for a Dedekind domain this is equivalent to the ideal class group being a torsion group (see also Theorem 2.2 of C. M. Reis and T. M. Viswanathan [A compactness property for prime ideals in Noetherian rings, Proc. Amer. Math. Soc. 25 (1970), 353–356]). We also show that every compactly packed ring R is coprimely packed. For characterizations of coprimely packed Prüfer domains from a different point of view see V. Erdoǧdu [Modules with locally linearly ordered distributive hulls, J. Pure Appl. Algebra 47 (1987), 119–130].