Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 4 2 2023 07 01 Certain modules with the Noetherian dimension 121 129 172737 10.30504/jims.2023.399248.1118 EN S. M. Javdannezhad Department of Science, Shahid Rajaee Teacher Training University, Tahran, Iran. N. Shirali Department of mathematics, Shahid chamran university of Ahvaz, Ahvaz, Iran. Journal Article 2023 05 27 ‎An $R$-Module $M$ with a small submodule $S$, ‎such that $\frac{M}{S}$ is Noetherian, ‎is called a $SN$-module‎. ‎In this paper‎, ‎we introduce the concept of $\alpha$-$SN$-modules‎, ‎for any ordinal $\alpha \geq 0$ ($SN$-modules are just $0$-$SN$-modules)‎. ‎Some of the basic results of $SN$-modules extended to $\alpha$-$SN$-modules‎. ‎It is shown that an $fs$-module $M$‎, ‎which is $\alpha$-$SN$‎, ‎has Noetherian dimension $\leq \alpha$‎. ‎In particular‎, ‎if $M$ is quotient finite-dimensional and all of its submodules are $\alpha$-$SN$‎, ‎then $M$ has Noetherian dimension $\leq \alpha$‎. ‎Furthermore‎, ‎the concepts of $qn$-submodules (a proper submodule $N$ of $M$ is called a $qn$-submodule if $\frac{M}{N}$ has Noetherian dimension) and $qn$-modules are introduced‎. ‎It is proved that if $M$ is quotient finite-dimensional and each of its submodules has at least a $qn$-submodule‎, ‎then $M$ has Noetherian dimension. Some other results are obtained too. https://jims.ims.ir/article_172737_40518551c96582a906707c7b63429958.pdf