# The length of Artinian modules with countable Noetherian dimension

Document Type: Research Paper

Authors

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

Abstract

‎It is shown that‎ ‎if $M$ is an Artinian module over a ring‎ ‎$R$‎, ‎then $M$ has Noetherian dimension $\alpha$‎, ‎where $\alpha$ is a countable ordinal number‎, ‎if and only if $\omega ^{\alpha }+2\leq \it{l}(M)\leq \omega ^{\alpha‎ +1}$, ‎where $\it{l}(M)$ is‎ ‎the length of $M$‎, ‎$i.e.,$ the least ordinal number such that the interval $[0‎, ‎\ \it{l}(M))$ cannot be embedded in the lattice of all submodules of $M$.

Keywords

Main Subjects

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### History

• Receive Date: 07 April 2016
• Revise Date: 26 July 2016
• Accept Date: 27 July 2016