On $α$-Short Modules - Davoudian, M.; Karamzadeh, O. A. S.; Shirali, N. (MATHEMATICA SCANDINAVICA, 2014)
We introduce and study the concept of $α$-short modules (a $0$-short module is just a short module, i.e., for each submodule $N$ of a module $M$, either $N$ or $\frac{M}{N}$ is Noetherian). Using this concept we extend some of the basic results of short modules to $α$-short modules. In particular, we show that if $M$ is an $α$-short module, where $α$ is a countable ordinal, then every submodule of $M$ is countably generated. We observe that if $M$ is an $α$-short module then the Noetherian dimension of $M$ is either $α$ or $α+1$. In particular, if $R$ is a semiprime ring, then $R$ is $α$-short as an $R$-module if and only if its Noetherian dimension is $α$.