The small condition for modules with Noetherian dimension

Shirali, Nasrin; Kavoosi Ghafi, Hooriya; Javdannezhad, Sayed Malek

doi:10.22103/jmmr.2024.23633.1670

The small condition for modules with Noetherian dimension

Document Type : Research Paper

Authors

Nasrin Shirali ¹

Hooriya Kavoosi Ghafi ¹

Sayed Malek Javdannezhad ²

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

² Department of Mathematics, Shahid Rajaee Teacher Training University, Tehran, Iran

10.22103/jmmr.2024.23633.1670

Abstract

A module $M$ with Noetherian dimension is said to satisfy the small condition, if for any small submodule $S$ of $M$ the Noetherian dimension of $S$ is strictly less than the Noetherian dimension of $M$. For an Artinian module $M$, this is equivalent to that $M$ is semisimple. In this article, we introduce and study this concept and observe some basic facts for modules with this condition. As a main result, it is shown that if $M$ is a module with finite hollow dimension which satisfies the small condition, then $\alpha \leq n-dim\, M\leq \alpha+1$, where $\alpha=\sup\{ n-dim\,S: S\ll M\}$. Furthermore, if $M$ is a module with Krull dimension and finite hollow dimension, then $\alpha \leq k-dim\, M\leq \alpha+1$, where $\alpha=\sup\{ k-dim\,S: S\ll M\}$. Also, we study the projective cover of modules satisfying the small condition or with finite hollow dimension

Keywords

Large condition

Small condition

Semiatomic modules

Projective cover

Subjects

Algebra

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