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.

Keywords: Nonfinitely generated modules, Krull dimension, DICC-modules, n.f.g-DICC modules

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