A compactness property for prime ideals in Noetherian rings

Abstract

A ring R is compactly packed by prime ideals if whenever an ideal / of R is contained in the union of a family of prime ideals of R, I is actually contained in one of the prime ideals of the family.It is shown that a commutative Noetherian ring is compactly packed if and only if every prime ideal is the radical of a principal ideal.For Dedekind domains this is equivalent to the torsion of the ideal class group and again to the existence of distinguished elements for the essential valuations.If a Noetherian ring R is compactly packed then Krull dim.R 5j 1.Also a Krull domain R is compactly packed if and only if it is a Dedekind domain with torsion ideal class group.Throughout R will be assumed to be a commutative ring with an identity element.The following property is well known [l, p. 8]: if an ideal I^UlZl Pi where the P,-are prime ideals of P then I^Pt for some i.This note discusses the situation when the union of any family of prime ideals of R is considered.We say that the ring P is compactly packed (by prime ideals of R) if the following holds: (*) if an ideal /CUaesP" where the Pa (aES)are prime ideals of P then I^Pa for some aES.p. 213].Since xEP, one of the P* say Px is necessarily contained in P.We have two cases.Either k = 1 in which case Pi$P or k> 1.In that case again Pcf1P2, for from PQP2, we will have PiCPCP2 contradicting the fact that Pi, P2 are isolated prime components of (x).Thus in either case, there exists a prime ideal Qx such that PQiQx and