Short modules and almost noetherian modules

Abstract

It is proved that, for any ring $R$, a right $R$-module $M$ has the property that, for every submodule $N$, either $N$ or $M/N$ is Noetherian if and only if $M$ contains submodules $K \supseteq L$ such that $M/K$ and $L$ are Noetherian and $K/L$ is almost Noetherian.