Primary Ideals and Prime Power Ideals

Abstract

This paper is concerned with the ideal theory of a commutative ring R. We say R has Property ( α ) if each primary ideal in R is a power of its (prime) radical; R is said to have Property ( δ ) provided every ideal in R is an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that if D is a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, then D is a Dedekind domain. It follows from this that if D has Property ( α ) and is Noetherian (in which case D has Property ( δ )), then D is Dedekind.