On the associated graded ring of a semigroup ring

Let (R, m) be a numerical semigroup ring. In this paper we study the properties of its associated graded ring G(m). In particular, we describe the (formula) for G(m) (where M is the homogeneous maximal ideal of G(m)) and we characterize when G(m) is Buchsbaum. Furthermore, we find the length of as a G(m)-module, when G(m) is Buchsbaum. In the 3-generated numerical semigroup case, we describe the (formula) in terms of the Apery set of the numerical semigroup associated to R. Finally, we improve two characterizations of the Cohen-Macaulayness and Gorensteinness of G(m) given in [2, 3], respectively. © 2011, Rocky Mountain Mathematics Consortium. All rights reserved.