Se p 20 02 Notes on the behavior of the Ratliff - Rush filtration

We establish new classes of Ratliff-Rush closed ideals and some pathological behavior of the Ratliff-Rush closure. In particular, Ratliff-Rush closure does not behave well under passage modulo superficial elements, taking powers of ideals, associated primes, leading term ideals, and the minimal number of generators. In contrast, the reduction number of the Ratliff-Rush filtration behaves better: it preserves some information on the reduction number of the ideal. Let I be an ideal in a Noetherian ring R. From the maximal condition it follows that there exist ideals Ĩ in R maximal with respect to the condition Ĩ n = I for all large n. Ratliff and Rush proved in [RR, Theorem 2.1], that if I is a regular ideal (i.e., it contains a nonzerodivisor), then there exists a unique largest such Ĩ , which can be presented in terms of I as follows: Ĩ := ⋃