Weakly uniserial dimension of modules

Abstract

Recall that a module is called weakly uniserial if its submodules are comparable regarding embedding. Weakly uniserial modules are a nontrivial generalization of uniserial modules. In this paper we define and study a new dimension, which measure how far a module deviates from being weakly uniserial. We call this dimension, weakly uniserial dimension. Also, we define and study monoartinian (mononoetherian) modules. We say that an $R$-module $M$ is monoartinian (mononoetherian) if in every descending (ascending) chain of submodules of $M$, except probably a finite number, each module in chain embedded in the next (previous) one. We show that a module has weakly uniserial dimension if and only if it is monoartonian.