A note on the hilbertfunction of a one-dimensional Cohen-Macaulay ring

Abstract

In this note we give examples of one-dimensional noetherian local integral domains with the property that the number of generators of the square of the maximal ideal is less than the embedding dimension. On the other hand we show that if $$(R,m)$$ is a local one-dimensional Cohen-Macaulay ring, then $$(m^n )$$ for all n such that μ $$(m^n )$$ < m(R). Here μ denotes the minimal number of generators of an ideal and m(R) the multiplicity of R. A similar statement was conjectured by D. Sally in [5], Our main result 2.1 generalizes Prop. 2.6. in [6].