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(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I=Q:\mathfrak{m}. Then the equality I²=QI holds true, if e(A)=2 and depth A>0. The assertion is no longer true, unless e(A)=2$. Counterexamples are given.