The small condition for modules with Noetherian dimension

Document Type : Research Paper

Authors

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

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

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

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Main Subjects

References

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Journal of Mahani Mathematical Research
Volume 14, Issue 1 - Serial Number 31
January 2025
Pages 327-343

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History

  • Receive Date: 15 June 2024
  • Revise Date: 07 September 2024
  • Accept Date: 18 October 2024

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