The small condition for modules with Noetherian dimension

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