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**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.

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.

Reviewer: Siamak Yassemi (Tehran)

### MSC:

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 |