On $AB5^*$ modules with Noetherian dimension

Document Type : Original Article

Authors

Nasrin Shirali ¹

Sayed Malek Javdannezhad ²

¹ Department of Mathematics‎, ‎Shahid Chamran University of Ahvaz‎, ‎Ahvaz‎, ‎Iran

² Department of Science‎, ‎Shahid Rajaee Teacher Training University‎: ‎Tehran‎, ‎Iran

10.22034/kjm.2024.382530.2750

Abstract

In this paper, we study the Noetherian dimension of sum of certain modules. It is proved that for any module $M$ which is an irredundant sum of submodules, each of which has Noetherian dimension $\leq \alpha$, if $M$ has finite spanning dimension ($fsd$-module, for short) or it is a weakly atomic module, then $n-\dim\,M \leq \alpha$. Here, by a weakly atomic module we mean a module $M$ for which every proper non-small submodule $N$, has Noetherian dimension strictly less than that of $M$. Also, it is proved that if $M$ is an $AB5^*$ module with Noetherian dimension and $\{N_i\}$ is a family of submodules of $M$ such that $n-\dim\,\frac{M}{N_i} \leq \alpha$, for each $i$, then $n-\dim\,\frac{M}{\cap N_i} \leq \alpha$. Using this, we give a structure theorem for $\alpha$-short modules in the category of $AB5^*$ and finally, we classify $\alpha$-short modules in this category.

Keywords

Noetherian dimension

Krull dimension

$AB5^*$ Modules

$\alpha$-short modules}

Subjects

• 16 Associative rings and algebras

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