Reductions and cores of ideals

Abstract

In 1954, Northcott and Rees introduced the notions of reduction of an ideal and basic ideal in Noetherian local rings with infinite residue field. In 1973-1975, Hays investigated these concepts in more general settings; namely, Noetherian rings and Prüfer domains. The core of an ideal is the intersection of all its reductions. This notion was initially introduced by Northcott and Rees (1954) and later systematically studied by Sally and Rees (1988). In 1995, Huneke and Swanson determined explicitly the core of integrally closed ideals in 2-dimensional regular Noetherian local rings. The goal of this MS Thesis is to study in detail the above concepts.