The goal of this paper is to create an algebraic structure based on single-valued neutrosophic sets. We present a novel approach to the neutrosophic sub-ring and ideal by combining the classical ring with neutrosophic sets. We also introduce and investigate some of the fundamental properties of the concepts. Finally, we show how to use a neutrosophic ideal to make a decision.

A module represents a fundamental and complicated algebraic structure associated with a particular binary operation in algebraic theory. This paper introduces a new class of neutrosophic sub-module and neutrosophic R-sub-module. We extend the basic definitions in this area for the first time. Various properties of a neutrosophic R-sub-module are studied in different classes of rings. Moreover, various definitions of direct product and homomorphism of neutrosophic R-sub-modules are discussed, and results are provided.

This article aims to introduce and explore the concept of neutrosophic ideals in the context of rings. Besides, we investigate how the property of regularity in a ring can be understood through the lens of neutrosophic ideals. We further present the concept of neutrosophic prime ideals and systematically identify all neutrosophic prime ideals in $\mathbb{Z}$.