ON α-SHORT MODULES

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 $\frac{\mathrm{M}}{\mathrm{N}}$ 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.