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.