Commutative rings whose prime ideals are radically perfect

Abstract

One of the interesting research topics in commutative algebra is to relate the number of generators of an ideal to its height. In [Arch. Math. 93, No. 3, 213--217 (2009; Zbl 1180.13014)] the first author of the present paper calls 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 \(I\). This is a generalization of the notion of set theoretic complete intersection of ideals in Noetherian rings to rings that need not be Noetherian. The main aim of the paper under review is to determine conditions on a ring \(R\) so that the prime ideals of \(R\) and also those of the polynomial rings \(R[X]\) over \(R\) are radically perfect. In many cases, it is shown that the condition of prime ideals of \(R\) or that of \(R[X]\) being radically perfect is equivalent to a form of the class group of \(R\) being torsion. This is a well-written paper with some interesting results.