Reducibility of parameter ideals in low powers of the maximal ideal

Abstract

A commutative noetherian local ring (R, m) is Gorenstein if and only if every parameter ideal of R is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer ` (depending on R) such that R is Gorenstein if and only if there exists an irreducible parameter ideal contained in m` . We give upper bounds for ` that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.