Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808812320220923On $\alpha$-parallel short modulesOn $\alpha$-parallel short modules4374471787610.22055/jamm.2022.41194.2053FASayed Malek JavdannezhadFaculty of Science, Shahid Rajaee Teacher Training University, Tehran, IranNasrin ShiraliِDepartment of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran0000-0002-9907-7352Maryam ShiraliِDepartment of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, IranSayedeh Fatemah MousavinasabِDepartment of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, IranJournal Article20220626An $R$-module $M$ is called $\alpha$-parallel short modules, if for each parallel submodule $N$ to $M$ either $\pndim\, N \leq \alpha$ or $\ndim\, \frac{M}{N}\leq\alpha$ and $\alpha$ is the least ordinal<br /><br />number with this property. Using this concept, we extend some of the basic results of $\alpha$-short modules<br /><br />to $\alpha$-parallel short modules.<br /><br />Also, we have studied the relationship between $\alpha$-parallel short modules and their parallel Noetherian dimension and we show that if $M$ is a $\alpha$-parallel short module, then $M$ has parallel Noetherian dimension and<br /><br />$\alpha\leq\pndim\, M\leq \alpha+1$. Furthermore, we prove that if $M$ is an $\alpha$-parallel short<br /><br />module with finite Goldie dimension, then $M$ has Noetherian dimension and $\alpha\leq\ndim\, M\leq\alpha+1$.An $R$-module $M$ is called $\alpha$-parallel short modules, if for each parallel submodule $N$ to $M$ either $\pndim\, N \leq \alpha$ or $\ndim\, \frac{M}{N}\leq\alpha$ and $\alpha$ is the least ordinal<br /><br />number with this property. Using this concept, we extend some of the basic results of $\alpha$-short modules<br /><br />to $\alpha$-parallel short modules.<br /><br />Also, we have studied the relationship between $\alpha$-parallel short modules and their parallel Noetherian dimension and we show that if $M$ is a $\alpha$-parallel short module, then $M$ has parallel Noetherian dimension and<br /><br />$\alpha\leq\pndim\, M\leq \alpha+1$. Furthermore, we prove that if $M$ is an $\alpha$-parallel short<br /><br />module with finite Goldie dimension, then $M$ has Noetherian dimension and $\alpha\leq\ndim\, M\leq\alpha+1$.https://jamm.scu.ac.ir/article_17876_fd16aa91922dd3d46065234f5c5cc928.pdf