The Noetherian dimension of modules versus their \alpha-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{\rm -dim}\, A, 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{\rm -dim}\, A\leq n{\rm -dim}\, A\leq sn{\rm -dim}\, A+۱. Last we introduce the concept of \alpha-small short modules, i.e., for each small submodule S of A, either n{\rm -dim}\, S \leqslant \alpha or sn{\rm -dim}\,\frac{A}{S}\leqslant\alpha and \alpha is the least ordinal number with this property and by using this concept, we extend some of the basic results of short modules to \alpha-small short modules. In particular, we prove that if A is an \alpha-small short module, then it has small Noetherian dimension and sn{\rm -dim}\, A=\alpha or sn{\rm -dim}\, A=\alpha+۱. Consequently, we show that if A is an \alpha-small short module with finite hollow dimension, then \alpha\leq n{\rm -dim}\, A\leq\alpha+۲.

Keywords:

Small Noetherian dimension , Hollow dimension , Small short modules , Small atomic module