Radically perfect prime ideals in polynomial rings

Abstract

Let \(R\) be a commutative ring, let rad\((I)\) denote the radical of the ideal \(I\subset R\), let \(S(I)\) be the set of all ideals \(J\subset R\) with rad\((J)\)=rad\((I)\), and let \(I^*\) be an ideal in \(S(I)\) with the least number of generators. The author calls an ideal \(I\) \textit{radically perfect} if the number of generators of \(I^*\) equals the height of \(I\) (usually such ideals are said to be set-theoretic complete intersections). The author considers Noetherian domains \(R\) of Krull dimension zero containing the field of rational numbers, and shows that every prime ideal of \(R[X]\) is radically perfect if and only if \(R\) is a Dedekind domain having a torsion class-group. Earlier he established this assuming additionally that \(R\) is normal [Proc. Am. Math. Soc. 132, 3467--3471 (2004; Zbl 1081.13001)]. He characterizes also finite dimensional Bézout domains \(R\) such that eery prime ideal of \(R[X]\) is radically perfect.