Quintasymptotic primes and four results of Schenzel
Let I, resp. S, denote an ideal, resp. a multiplicatively closed set of a Noetherian ring R. By \(T_ 1, T_ 2, T_ 3\) denote the topologies defined by the filtrations \(\{I^ nR_ S\cap R\}\), \(\{I^ n\}\), \(\{(I^ n)_ a\}\), \(n\geq 1\). Here \(I_ a\) denotes the integral closure of I. In the reviewer's paper [Math. Nachr. 129, 123-148 (1986; Zbl 0606.13001)], among others, it is solved when \(T_ 1\) is equivalent resp. linearly equivalent to either of \(T_ 2\) or \(T_ 3\). In the paper under review the author gives a concise proof of these results. In fact, the arguments follow more or less the same line based on a variation of a theorem of Chevalley [\textit{O. Zariski} and \textit{P. Samuel}, ``Commutative algebra'', Vol. II (Princeton 1960; Zbl 0121.278); reprint 1976] and its application to Rees rings. In contrast to the reviewer's paper cited above the author uses consequently the notion of several types of asymptotic prime divisors in order to describe the main results. Furthermore he considers new relations among the various types of asymptotic prime divisors resp. differences between them by examples. Finally he continues with the study of so-called quintasymptotic prime divisors begun by the reviewer (loc. cit.) under a different notation.
scientific article; zbMATH DE number 4043969