Certain modules with the Noetherian dimension

Document Type : Dedicated to Prof. O. A. S. Karamzadeh


1 Department of Science, Shahid Rajaee Teacher Training University, Tahran, Iran.

2 Department of mathematics, Shahid chamran university of Ahvaz, Ahvaz, Iran.


‎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.


Main Subjects

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