Euclidean-like characterizations of Dedekind, Krull, and factorial domains (Q1328372)

Let \(A\) be an integral domain. The author defines a function \(N\) from the quotient field \(K\) of \(A\), the monoid \(M(A)\) of ideals of \(A\), the partially ordered commutative monoid \(D(A)\) of divisorial ideals of \(A\) with values in the field of rational numbers or the multiplicative group of positive rationals. He uses this function \(N\), which behaves like a norm in a number field, to give necessary and sufficient conditions for \(A\) to be (i) a principal ideal domain, (ii) a Dedekind domain, (iii) a Krull domain, and (iv) a factorial domain.