Results on weakly uniserial modules

In this paper, we study weakly uniserial modules, a concept recently introduced by Moradzadeh-Dehkordi et al., which extends the notion of uniserial modules. A module $M$ is said to be weakly uniserial if for any submodules $N$ and $L$ of $M$, there exists a monomorphism $N \rightarrowtail L$ or $L \rightarrowtail N$. Our analysis explores the relationship between weakly uniserial modules and classical notions in ring and module theory, including preradicals, socle series, the singular submodule, injective hulls, and $V$-rings. In addition, we present further statements complementing the characterization provided by the aforementioned authors concerning rings over which every module is weakly uniserial. Finally, by using monomorphisms, we resolve the Schröder–Bernstein problem within the class of isoartinian modules.