THE NOETHERIAN DIMENSION OF MODULES VERSUS THEIR α-SMALL SHORTNESS

In this article, we first consider concept of small Noetherian dimension of a module, which is dual to the small krull dimension, denoted by sn-dimA, and defined to be the codeviation of the poset of the small submodules of A. We prove that if an R-module A with finite hollow dimension, has small Noetherian dimension, then A has Noetherian dimension and sn-dimA ≤ n-dimA ≤ sn-dimA+1. Last we introduce the concept of α-small short modules, i.e., for each small submodule S of A, either n-dim S ⩽ α or sn-dim A S ⩽ α and α is the least ordinal number with this property and by using this concept, we extend some of the basic results of short modules to α-small short modules. In particular, we prove that if A is an α-small short module, then it has small Noetherian dimension and sn-dimA = α or sn-dimA = α + 1. Consequently, we show that if A is an α-small short module with finite hollow dimension, then α ≤ n-dimA ≤ α + 2.