On n-submodules and G.n-submodules

We investigate some properties of n-submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an n-submodule. Also, we show that if M is a finitely generated R-module and AnnR(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt {{\rm{An}}{{\rm{n}}_R}\left(M \right)} $$\end{document} is a prime ideal of R, then M has n-submodule. Moreover, we define the notion of G.n-submodule, which is a generalization of the notion of n-submodule. We find some characterizations of G.n-submodules and we examine the way the aforementioned notions are related to each other.