The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two

Abstract

Let $A$ be a Buchsbaum local ring with the maximal ideal $\mathfrak{m}$ and let $e\left(A\right)$ denote the multiplicity of $A$. Let $Q$ be a parameter ideal in $A$ and put $I=Q$ : $\mathfrak{m}$. Then the equality $I^2=QI$ holds true, if $e\left(A\right)=2$ and $\mathrm{depth}A>0$. The assertion is no longer true, unless $e\left(A\right)=2$. Counterexamples are given.