On finiteness conditions of modular lattices

Abstract

In [Gd] Goodearl proved that if for every essential submodule N of a module M M/N is a Noetherian module, then the module M/SocM is Noetherian. Then in [AS], Al-Khazzi and Smith got that if every small submodule of a module M is Artinian then so is the Jacobson radical 𝔍(M) of M. These results are dual to each other in the lattice theory sense ( recall that is essential submodule of M}). However the proofs in [AS] and [Gd] are not dual at all. Hence it is natural to ask whether there is a common proof of the both results. The best it would be to extend one of these results to complete modular lattices or to a satisfactory subclass of such lattices. Then to get any of the results it would be enough to take the lattice of submodules of a module or its dual. Attempts to find such an extension inspired our studies in this paper. We did not settle the general problem but obtained such an extension in cases of complete modular lattices which are upper continuous, lower continuous or distributive. Moreover we got some general results on complete modular lattices which, applied to modules, give uniform proofs of several known results on dimensions of modules