The Noetherian dimension of modules versus their $\alpha$-small shortness

Document Type : Research Paper


Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran


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+1$. 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 $sn{\rm -dim}\, A=\alpha+1$. Consequently, we show that if $A$ is an $\alpha$-small short module with finite hollow dimension, then $\alpha\leq n{\rm -dim}\, A\leq\alpha+2$.


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