REDUCTIONS OF IDEALS IN PRÜFER RINGS

Abstract

Let $R$ be a ring and $I$ a proper ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $J{I}^n=I^{n+1}$ for some positive integer $n$; and $I$ is called basic if it has no proper reductions. The notion of reduction was introduced by Northcott and Rees with the initial purpose to contribute to the analytic theory of ideals in Noetherian (local) rings via reductions.

Two well-known results, due to Hays, assert that an integral domain is Prüfer if and only if every finitely generated ideal is basic, and it is one-dimensional Prüfer if and only if every ideal is basic. This paper investigates reductions of ideals in the family of Prüfer rings, with the aim to recover and generalize Hays’ results to classes of rings with zero-divisors subject to various Prüfer conditions.