Gorenstein rings and irreducible parameter ideals

Abstract

Given a Noetherian local ring ( R , m ) (R,m) it is shown that there exists an integer ℓ \ell such that R R is Gorenstein if and only if some system of parameters contained in m ℓ m^{\ell } generates an irreducible ideal. We obtain as a corollary that R R is Gorenstein if and only if every power of the maximal ideal contains an irreducible parameter ideal.