$\phi$-$\delta$-$S$-primary submodule

TL;DR

This paper introduces and studies $$-$ $-$S$-primary submodules, generalizing existing concepts of primary and prime submodules, and establishes a correspondence between these submodules in modules and their quotients.

Contribution

It extends the theory of $$-$ $-primary submodules to a more general framework and explores their properties and relationships within module theory.

Findings

Established properties of $$-$ $-$S$-primary submodules.

Proved a one-to-one correspondence with submodules of quotient modules.

Abstract

In this paper, we introduce and investigate some properties of $\phi$-$\delta$-$S$-primary submodules, which is a generalization of the $\phi$-$\delta$-primary submodules and prime submodules in general. We extend a number of main results about $\phi$-$\delta$-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 $\phi$-$\delta$-$S$-primary submodules of an $R$-module $M$ containing $K$ and the set of all $\phi$-$\delta$-$S$-primary submodules of $\frac{M}{K}$.