The first section of this paper is
devoted to proving the following theorem. Let D be an integral domain with identity.
Let 𝒫 be the set of prime powers of D, 𝒱 the set of valuation ideals of D, and let k
be the quotient field of D. 𝒱⊆𝒫 if and only if the following conditions hold: (i) Each
prime ideal P of D defines a P-adic valuation in the sense of van der Waerden, and
(ii) every valuation of k finite on D is isomorphic to a P-adic valuation for some
P.
The second section considers three additional sets of ideals; the set 𝒬 of primary
ideals, the set 𝒮 of semi-primary ideals, and the set 𝒜 of ideals A such that the
complement of some prime ideal is prime to A.
