ON THE NOETHERIAN DIMENSION OF ARTINIAN MODULES WITH HOMOGENEOUS UNISERIAL DIMENSION

In this article, we first show that non-Noetherian Artinian uniserial modules over commutative rings, duo rings, finite R-algebras and right Noetherian rings are 1-atomic exactly like Z(p infinity) . Consequently, we show that if R is a right duo (or, a right Noetherian) ring, then the Noetherian dimension of an Artinian module with homogeneous uniserial dimension is less than or equal to 1. In particular, if A is a quotient finite dimensional R-module with homogeneous uniserial dimension, where R is a locally Noetherian (or, a Noetherian duo) ring, then n-dim A <= 1. We also show that the Krull dimension of Noetherian modules is bounded by the uniserial dimension of these modules. Moreover, we introduce the concept of qu-uniserial modules and by using this concept, we observe that if A is an Artinian R-module, such that any of its submodules is qu-uniserial, where R is a right duo (or, a right Noetherian) ring, then n-dim A <= 1.