Strongly Bounded Rings

Abstract

Despite the apparent power of tight closure techniques, the tight closure operation itself is quite difficult to handle in practice. For example, it is generally difficult to find the tight closure of an arbitrary ideal. Also, it is not known whether tight closure behaves well under localization. We y1 Ž y1 . would like to know whether it is true that I*W R s IW R * where I is an ideal of a ring R, W is an arbitrary multiplicative system, and I* denotes the tight closure of I. It is not even known that if all of the ideals of R are tightly closed, then all of the ideals of R are tightly closed.