On $\alpha$-Short Modules

  • M. Davoudian
  • O. A. S. Karamzadeh
  • N. Shirali


We introduce and study the concept of $\alpha$-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 $\alpha$-short modules. In particular, we show that if $M$ is an $\alpha$-short module, where $\alpha$ is a countable ordinal, then every submodule of $M$ is countably generated. We observe that if $M$ is an $\alpha$-short module then the Noetherian dimension of $M$ is either $\alpha$ or $\alpha+1$. In particular, if $R$ is a semiprime ring, then $R$ is $\alpha$-short as an $R$-module if and only if its Noetherian dimension is $\alpha$.
How to Cite
Davoudian, M., Karamzadeh, O. A. S., & Shirali, N. (2014). On $\alpha$-Short Modules. MATHEMATICA SCANDINAVICA, 114(1), 26-37. https://doi.org/10.7146/math.scand.a-16638