F. G. NOETHERIAN MODULES AND F. G. ARTINIAN MODULES

In this paper, we introduced the notions f.g. Noetherian and f.g. Artinian modules. Also we nd some characterizations of them. We show that an R-module M is f.g. Noetherian (Artinian) if and only if every nonempty subset of its nitely generatedsubmodules has a maximal (minimal) element. Also we show that for maximal ideals m1; :::;mn of R, if M is a nitely generated R-module such that m1:::mnM = 0, then M is f.g. Noetherian if and only if M is f.g. Artinian.

Keywords:

Artinian module , f.g. Artinian module , f.g. Noetherian module , Finitely generated submodule , Noetherian module