Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16124220230701Certain modules with the Noetherian dimension12112917273710.30504/jims.2023.399248.1118ENS. M.JavdannezhadDepartment of Science, Shahid Rajaee Teacher Training University, Tahran, Iran.N.ShiraliDepartment of mathematics, Shahid chamran university of Ahvaz, Ahvaz, Iran.Journal Article20230527An $R$-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_1e16d4a46fb4f5da915b017734ac306c.pdf