Abstract. In this note we prove a structure theorem for a right Noetherian ring R with krull dimension one having a right Artinian quotient ring and, moreover, that has a finitely generated, faithful, critical right module U with krull dimension U equal to one. We end this note with some examples that clarify certain features of this theorem.
Notation and terminologyLet R be a ring and M a right .R-module. We denote the right krull dimension of M or R by \M\ or \R\. For basic properties of krull dimension, the reader is referred to [6]. A module with krull dimension is said to be akrull homogeneous or simply krull homogeneous in the case where \M\ = a, if |TV| = \M\, for every nonzero submodule N of M. The ring R is said to be krull homogeneous if the module Rr is krull homogeneous. For the definition of Primary module or Primary ring and various results about Prime ideals associated to a module we refer the reader to Gordon [4]. If A is a ring then its nilradical will be denoted by N(A). The set of regular elements of A is denoted by C{0) and C(B) denotes the set of elements of A that are regular modulo B where B is an ideal of A . If S,■■, i £ I, are subrings of a ring R then their internal direct sum is denoted by 4-,G/ Sj. A ring R is said to be nonsingular if the right singular ideal of R is zero. The set of associated prime ideals of a right module M is denoted by Ass(M). As usual rings are with identity and modules are unitary.Main theorem Proposition 1. Let R be a right Noetherian ring with M a faithful, finitely generated, critical module over R such that \M\ = \R\. Then R is a krull homogeneous P-primary ring, where P is the associated prime of M. Furthermore, if I is an essential right ideal of R then \R/I\ < \R\.