Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808812320220923On $\alpha$-parallel short modulesOn $\alpha$-parallel short modules4374471787610.22055/jamm.2022.41194.2053FASayed MalekJavdannezhadFaculty of Science, Shahid Rajaee Teacher Training University, Tehran, IranNasrinShiraliِDepartment of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran0000-0002-9907-7352MaryamShiraliِDepartment of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, IranSayedeh FatemahMousavinasabِDepartment 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