Abstract
We call an ideal I of a commutative ring R radically perfect if among the ideals of R whose radical is equal to the radical of I the one with the least number of generators has this number of generators equal to the height of I. Let R be a Noetherian integral domain of Krull dimension one containing a field of characteristic zero. Then each prime ideal of the polynomial ring R[X] is radically perfect if and only if R is a Dedekind domain with torsion ideal class group. We also show that over a finite dimensional Bézout domain R, the polynomial ring R[X] has the property that each prime ideal of it is radically perfect if and only if R is of dimension one and each prime ideal of R is the radical of a principal ideal.
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This research is supported in part by TUBITAK, Grant no: 107T312.
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Erdoğdu, V. Radically perfect prime ideals in polynomial rings. Arch. Math. 93, 213–217 (2009). https://doi.org/10.1007/s00013-009-0036-1
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DOI: https://doi.org/10.1007/s00013-009-0036-1