On Graded s-Prime Submodules

TL;DR

This paper introduces graded s-prime submodules as a generalization of graded prime submodules, exploring their properties under various module operations and establishing their existence in graded-Noetherian modules.

Contribution

It defines graded s-prime submodules, studies their behavior under homomorphisms and localization, and proves their existence in certain classes of graded modules.

Findings

Existence of graded s-prime submodules in graded-Noetherian modules.

Conditions for existence in general graded modules.

Results for specific gradings such as $ extbf{Z}$, finite groups, and crossed products.

Abstract

In this article, we introduce the concepts of graded $s$-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $s$-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of grading by $\mathbb{Z}$, a finite group, or a polycyclic-by-finite group, in addition to crossed product grading.