$\varphi$-$\delta$-$S$-primary submodule

Abstract

In this paper, we introduce and investigate some properties of $\varphi$-$\delta$-$S$-primary submodules, which is a generalization of the $\varphi$-$\delta$-primary submodules and prime submodules in general. We extend a number of main results about $\varphi$-$\delta$-primary submodule and prime submodule into the general framework. Moreover, we show that there is an one-to-one correspondence between the set of all $\varphi$-$\delta$-$S$-primary submodules of an $R$-module $M$ containing $K$ and the set of all $\varphi$-$\delta$-$S$-primary submodules of $\frac{M}{K}$.