DIMENSION, MULTIPLICITY AND HILBERT FUNCTION OF ARTINIAN MODULES∗

Abstract

Throughout this paper, R denotes a commutative ring and M an Artinian R-module. The concept of Krull dimension for Artinian modules M was introduced first by R. N. Roberts in [10]. D. Kirby in [4] changed the terminology of Roberts and refered to Noetherian dimension (N-dim)to avoidconfusion with Krull dimension of Noetherian modules. Then, if (R,m) a quasi-local ring (i.e. m is the unique maximal ideal of R, but R is not necessarily Noetherian), Roberts proved that