Radical perfectness of prime ideals in certain integral domains. (English) Zbl 1364.13005

Let \(D\) be an integral domain. An ideal \(I\) of \(D\) is called radically perfect if the height of \(I\) is equal to the infimum of the number of generators of ideals of \(D\) whose radical is equal to the radical of \(I\). This generalizes of the notion of a set-theoretic complete intersection to non-Noetherian rings. In addition, \(D\) is called \(t\)-compactly packed if and only if every prime \(t\)-ideal of \(D\) is the radical of a principal ideal. Let \(T(D)\) be the set of \(t\)-invertible fractional \(t\)-ideals of \(D\). Then \(T(D)\) is an abelian group under the \(t\)-multiplication \(I*J = (IJ)t\). Hence, if we let \(P(D)\) be the subgroup of nonzero principal fractional ideals of \(T(D)\), then \(Cl(D) := T(D)/P(D)\), called the (\(t\)-)class group of \(D\), is an abelian group. The following theorem is one of the main result in the present paper:
Theorem. If \(D\) is a UMT-domain, then \(D[X]\) is \(t\)-compactly packed if and only if \(D\) is a \(t\)-compactly packed domain with \(Cl(D[X])\) torsion. In addition, the authors introduce the concepts of Serre’s conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.


13A15 Ideals and multiplicative ideal theory in commutative rings
13E99 Chain conditions, finiteness conditions in commutative ring theory
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13G05 Integral domains
Full Text: DOI Euclid