بررسی مدول های α- کوتاه موازی

در اﯾﻦ ﻣﻘﺎﻟﻪ ﻣﺪولﻫﺎی αﮐﻮﺗﺎه ﻣﻮازی را ﺗﻌﺮﯾﻒ ﻣﯽﮐﻨﯿﻢ و وﯾﮋﮔﯽﻫﺎی اﺳﺎﺳﯽ ﻣﺪولﻫﺎی ﮐﻮﺗﺎهرا ﺑﺮای ﻣﺪولﻫﺎی αﮐﻮﺗﺎه ﻣﻮازی ﺗﻌﻤﯿﻢ ﻣﯽدﻫﯿﻢ. ﻫﻤﭽﻨﯿﻦ ارﺗﺒﺎط وﯾﮋﮔﯽ αﮐﻮﺗﺎه ﻣﻮازی ﺑﺎ ﺑﻌﺪ ﻧﻮﯾﺘﺮیﻣﻮازی ﻣﺪول را ﺑﺮرﺳﯽ ﻣﯽﮐﻨﯿﻢ و ﻧﺸﺎن ﻣﯽدﻫﯿﻢ ﮐﻪ اﮔﺮ m ﯾﮏ ﻣﺪول αﮐﻮﺗﺎه ﻣﻮازی ﺑﺎﺷﺪ، آنﮔﺎهm دارای ﺑﻌﺪ ﻧﻮﯾﺘﺮی ﻣﻮازی اﺳﺖ و ١ + α ⩽ ndim m ⩽ α. ﺑﻪ وﯾﮋه ﻧﺘﯿﺠﻪ ﻣﯽ ﮔﯿﺮﯾﻢ ﮐﻪﮐﻪ اﮔﺮ m ﯾﮏ ﻣﺪول αﮐﻮﺗﺎه ﻣﻮازی ﺑﺎ ﺑﻌﺪ ﮔﻠﺪی ﻣﺘﻨﺎﻫﯽ ﺑﺎﺷﺪ، آنﮔﺎه m دارای ﺑﻌﺪ ﻧﻮﯾﺘﺮی اﺳﺖ وα ≤ ndimm ≤ α + ١.

On $alpha$-parallel short modules

An $R$module $M$ is called $alpha$parallel short modules, if for each parallel submodule $N$ to $M$ either $pndim, N leq alpha$ or $ndim, frac{M}{N}leqalpha$ and $alpha$ is the least ordinalnumber with this property. Using this concept, we extend some of the basic results of $alpha$short modulesto $alpha$parallel short modules.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$alphaleqpndim, Mleq alpha+1$. Furthermore, we prove that if $M$ is an $alpha$parallel shortmodule with finite Goldie dimension, then $M$ has Noetherian dimension and $alphaleqndim, Mleqalpha+1$.