Modules satisfying double chain condition on nonfinitely generated submodules have Krull dimension

Abstract

We prove the result in the title. We study submodules $N$ of a module $M$ such that whenever $\frac{M}{N}$ satisfies the double infinite chain condition so does $M$.Moreover, we observe that an $\alpha $-atomic module, where $\alpha\geq 2$ is an ordinal number, satisfies the previous chain if and only if it satisfies the descending chain condition on nonfinitely generated submodules.