On Finitely Generated Neutrosophic Modules with Finite Value Distribution

Abstract

This paper presents the novel framework of neutrosophic modules, an algebraic structure that arises by superimposing neutrosophic sets on classical module theory. The core of this study lies in the investigation of the structural symmetry between the axioms of a module and the trivalent nature of the neutrosophic set. We define a new class of modules based on the neutrosophic set. In addition, this study establishes and examines the categorical structure corresponding to neutrosophic $\mathfrak{R}$-modules. Furthermore, it presents the procedures by which finitely generated variants of these modules can be formulated and studied. Partial characterizations are given for a particular type in which the distribution of neutrosophic values remains within a finite set.

1. Introduction

The notion of a neutrosophic set was first introduced by Smarandache [1,2]. In this framework, every element of a universe of discourse X is characterized by three independent functions: the truth-membership function $\alpha$, the indeterminacy function $\beta$, and the falsity (or non-membership) function $\zeta$. These three functions are defined on X and may take values in $[0,1]$, without any restriction on their sum. Such a formulation allows one to model situations where uncertainty, inconsistency, or incomplete information prevents a precise evaluation.

In 2004, Kandaswamy and Smarandache [3] introduced a class of neutrosophic structures that rely exclusively on the indeterminacy component I, and they referred to these as I-neutrosophic algebraic structures.

In their construction, the underlying sets are built using neutrosophic numbers of the form $a+bI$, where $a,b\in \mathbb{R}$ (or in some ring/field under consideration), and I represents the indeterminacy element.

Due to the wide range of applications in decision science, computer engineering, and applied mathematics, researchers have emphasized the importance of redefining and generalizing algebraic operations in the context of neutrosophic sets [1,4,5]. Consequently, the study of algebraic structures in this framework cannot rely on a single approach; instead, multiple methods are required, each tailored to the specific application context. For this reason, several papers have presented different perspectives for the study of algebraic structures based on neutrosophic sets. For example, Çetkin et al. [6,7,8] have proposed new approaches for analyzing neutrosophic algebraic systems, while Elrawy et al. [9,10] have developed alternative formulations that extend classical algebraic notions to the neutrosophic setting. Overall, this work enriches the theoretical foundations of the field and opens up promising avenues for practical application in mathematics, computer science, and engineering.

The present research is devoted to establishing a comprehensive framework for studying modules in the context of neutrosophic sets. Drawing inspiration from the inherent triadic symmetry that underlies neutrosophic theory, this work extends the concept into the algebraic domain to construct and analyze a new class of neutrosophic modules. The primary objectives are to formulate the basic definitions and properties of these modules, to investigate the nature of their finitely generated forms, and to describe a particular subclass characterized by neutrosophic valuations restricted to a finite range. By achieving these goals, this study aims to enrich the theoretical foundations of algebraic systems influenced by neutrosophic logic and to open new directions for both mathematical analysis and potential applications in decision-making and information systems.

The remainder of this paper is organized as follows. The basic definitions and preliminaries are presented in Section 2. In Section 3, we introduce a new class of modules in the framework of neutrosophic sets, analyze their finitely generated forms, and discuss several related properties. Finally, Section 4 provides concluding remarks and highlights the main contributions of this study.

2. Preliminaries

The following concepts serve as preliminary tools that will be utilized in later parts of this work.

Definition 1

([11]). A category $\mathfrak{C}$ is a mathematical structure comprising:

• A class of objects, denoted $Obj\left(\mathfrak{C}\right)$.
• For every ordered pair of objects $(A,B)$, a class $Ho{m}_{\mathcal{C}}(A,B)$ of morphisms from A to B. A morphism f with domain A and co-domain B is written as $f:A\to B$.
• For every triple of objects $(A,B,C)$, a binary operation called composition:

  $$\circ :Ho{m}_{\mathcal{C}}(B,C)\times Ho{m}_{\mathcal{C}}(A,B)\to Ho{m}_{\mathcal{C}}(A,C),$$

  which takes a pair $(g,f)$ to the composite morphism $g\circ f$.

These morphisms are required to satisfy two fundamental axioms: associativity and identity.

Definition 2

([12]). Let $\mathfrak{R}$ be a ring with multiplicative identity. An $\mathfrak{R}$-module $\mathfrak{F}$ is termed a free module if there exists a subset $\mathfrak{E}\subseteq \mathfrak{F}$ that satisfies the following two conditions:

(1)

$\mathfrak{E}$ spans $\mathfrak{F}$.

(2)

Linear independence.

Such a set $\mathfrak{E}$ is called a basis for $\mathfrak{F}$. The module $\mathfrak{F}$ is then isomorphic to the direct sum of copies of $\mathfrak{R}$, indexed by the following basis $\mathfrak{E}$:

$$\mathfrak{F}\cong \underset{e\in \mathfrak{E}}{⨁}\mathfrak{R}.$$

This structure implies that free modules are characterized by a universal property: any map from the basis $\mathfrak{E}$ to another $\mathfrak{R}$-module $\mathfrak{M}$ can be uniquely extended to a module homomorphism from $\mathfrak{F}$ to $\mathfrak{M}$.

Definition 3

([4]). A neutrosophic set Ψ over $\mathbb{U}$ is described by

$$\Psi =\left\{<⌜,\mu (⌜),\gamma (⌜),\zeta (⌜)>\mid ⌜\in \mathbb{U}\right\},$$

where $\mu ,\gamma ,\zeta :\mathbb{U}⟶[0,1]$ satisfy the condition

$$0\le \mu (⌜)+\gamma (⌜)+\zeta (⌜)\le 3,\forall ⌜\in \mathbb{U}.$$

We now present the notion of a neutrosophic sub-module within the framework of modules.

Definition 4

([10]). Let M be a module. A collection of the form

$${\mathcal{M}}_M=\left\{<⌜,{\mu }_M(⌜),{\gamma }_M(⌜),{\zeta }_M(⌜)>\mid ⌜\in M\right\},$$

is said to be a neutrosophic $\mathfrak{R}$-sub-module of M provided that the functions

$${\mu }_M,{\gamma }_M,{\zeta }_M:M\to [0,1]$$

satisfy the following conditions for every $⌜,⌝\in M$ and for all $\ell \in \mathfrak{R}$:

1. 

For the sum of two elements

$$\begin{array}{cc}\hfill {\mu }_M(⌜+⌝)& \ge min\{{\mu }_M(⌜),{\mu }_M(⌝)\},\hfill \\ \hfill {\gamma }_M(⌜+⌝)& \le max\{{\gamma }_M(⌜),{\gamma }_M(⌝)\},\hfill \\ \hfill {\zeta }_M(⌜+⌝)& \le max\{{\zeta }_M(⌜),{\zeta }_M(⌝)\}.\hfill \end{array}$$

2. 

For scalar multiplication

$$\begin{array}{cc}\hfill {\mu }_M(\ell ⌜)& \ge min\{{\mu }_M(\ell ),{\mu }_M(⌝)\}={\mu }_M(⌜),\hfill \\ \hfill {\gamma }_M(\ell ⌜)& \le max\{{\gamma }_M(\ell ),{\gamma }_M(⌝)\}={\gamma }_M(⌜),\hfill \\ \hfill {\zeta }_M(\ell ⌜)& \le max\{{\zeta }_M(\ell ),{\zeta }_M(⌝)\}={\zeta }_M(⌜),\hfill \end{array}$$

where ${\mu }_M(\ell )=1$, ${\gamma }_M(\ell )=0$ and ${\zeta }_M(\ell )=0$ for every scalar $\ell \in \mathfrak{R}$.

3. 

For the zero element

$${\mu }_M\left(0\right)=1,{\gamma }_M\left(0\right)=0,{\zeta }_M\left(0\right)=0.$$

Definition 5

([9]). Let $\mathcal{M}$ be a neutrosophic subset of $\mathcal{N}$. For a fixed $\tau \in [0,1]$, we define the set

$${\mathcal{M}}_{\tau }=\left\{\langle ⌜,\mu (⌜),\gamma (⌜),\zeta (⌜)\rangle :⌜\in \mathbb{U},\mu (⌜)\ge \tau ,\gamma (⌜)\le \tau ,\zeta (⌜)\le \tau \right\}.$$

This collection is referred to as the $\sigma$-cut (or level set) of the neutrosophic subset $\mathcal{M}$.

3. Main Result

3.1. The Category of Neutrosophic Modules

Assume that $\mathfrak{R}$ is a classical ring and let M and N be denoted as left (or right) $\mathfrak{R}$-modules. Consider the set ${\mathcal{P}}_M=\{<⌜,{\alpha }_1(⌜),{\beta }_1(⌜),{\zeta }_1(⌜)>\mid ⌜\in M\}$, and let ${\mathcal{S}}_N=\{<⌝,{\alpha }_2(⌝),{\beta }_2(⌝),{\zeta }_2(⌝)>\mid ⌝\in N\}$ represent an arbitrary neutrosophic left (resp., right) $\mathfrak{R}$-module.

Definition 6.

Let ${\mathcal{P}}_M$ and ${\mathcal{S}}_N$ be any neutrosophic left (resp., right) $\mathfrak{R}$-modules. A neutrosophic $\mathfrak{R}$-morphism $\overline{\mathcal{F}}:{\mathcal{P}}_M\to {\mathcal{S}}_N$ is defined to fulfill the following criteria:

1. 

The function $\mathcal{F}:M\to N$ is a morphism of $\mathfrak{R}$-modules.

2. 

For every $⌜\in M$, it holds that ${\alpha }_2\left(\mathcal{F}(⌜)\right)\ge {\alpha }_1(⌜)$, ${\beta }_2\left(\mathcal{F}(⌜)\right)\le {\beta }_1(⌜)$ and ${\zeta }_2\left(\mathcal{F}(⌜)\right)\le {\zeta }_1(⌜)$.

Lemma 1.

Let $Hom({\mathcal{P}}_M,{\mathcal{S}}_N)$ denote the collection of all neutrosophic $\mathfrak{R}$-linear transformations from ${\mathcal{P}}_M$ to ${\mathcal{S}}_N$. This set forms an abelian group under point-wise addition. Furthermore, if the ring $\mathfrak{R}$ is commutative, then $Hom({\mathcal{P}}_M,{\mathcal{S}}_N)$ naturally acquires the structure of a left (resp., right) $\mathfrak{R}$-module.

Proof. 

Since

$$\begin{array}{cc}\hfill {\alpha }_2\left(0(⌜)\right)& ={\alpha }_2\left(0\right)=1\ge {\alpha }_1(⌜),\hfill \\ \hfill {\beta }_2\left(0(⌜)\right)& ={\beta }_2\left(0\right)=0\le {\beta }_1(⌜),\hfill \\ \hfill {\zeta }_2\left(0(⌜)\right)& ={\zeta }_2\left(0\right)=0\le {\zeta }_1(⌜),\hfill \end{array}$$

for all $⌜\in M$, one can define a neutrosophic $\mathfrak{R}$-map $\overline{0}\mid {\mathcal{P}}_M\to {\mathcal{S}}_N$.

Consider any two morphisms $\overline{\mathcal{F}},\overline{\mathcal{H}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$. Observe the following:

$$\begin{array}{cc}\hfill {\alpha }_2\left((\mathcal{F}+\mathcal{H})(⌜)\right)& ={\alpha }_2(\mathcal{F}(⌜)+\mathcal{H}(⌜))\hfill \\ & \ge min\{{\alpha }_2\left(\mathcal{F}(⌜)\right),{\alpha }_2\left(\mathcal{H}(⌜)\right)\}\hfill \\ & \ge min\{{\alpha }_1(⌜),{\alpha }_1(⌜)\}={\alpha }_1(⌜).\hfill \\ \hfill {\beta }_2\left((\mathcal{F}+\mathcal{H})(⌜)\right)& ={\beta }_2(\mathcal{F}(⌜)+\mathcal{H}(⌜))\hfill \\ & \le max\{{\beta }_2\left(\mathcal{F}(⌜)\right),{\beta }_2\left(\mathcal{H}(⌜)\right)\}\hfill \\ & \le max\{{\beta }_1(⌜),{\beta }_1(⌜)\}={\beta }_1(⌜).\hfill \\ \hfill {\zeta }_2\left((\mathcal{F}+\mathcal{H})(⌜)\right)& ={\zeta }_2(\mathcal{F}(⌜)+\mathcal{H}(⌜))\hfill \\ & \le max\{{\zeta }_2\left(\mathcal{F}(⌜)\right),{\zeta }_2\left(\mathcal{H}(⌜)\right)\}\hfill \\ & \le max\{{\zeta }_1(⌜),{\zeta }_1(⌜)\}={\zeta }_1(⌜).\hfill \end{array}$$

Thus, $\overline{\mathcal{F}}+\overline{\mathcal{H}}=\overline{\mathcal{F}+\mathcal{H}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$. This addition is clearly both commutative and associative.

Define negation for any $\overline{\mathcal{F}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ by

$$(-\overline{\mathcal{F}})(⌜)=-\overline{\mathcal{F}}(⌜),\forall ⌜\in M.$$

We confirm that this is valid, since

$$\begin{array}{cc}\hfill {\alpha }_2\left((-\overline{\mathcal{F}})(⌜)\right)& ={\alpha }_2(-\overline{\mathcal{F}}(⌜))={\alpha }_2\left(\overline{\mathcal{F}}(⌜)\right)\ge {\alpha }_1(⌜),\hfill \\ \hfill {\beta }_2\left((-\overline{\mathcal{F}})(⌜)\right)& ={\beta }_2(-\overline{\mathcal{F}}(⌜))={\beta }_2\left(\overline{\mathcal{F}}(⌜)\right)\le {\beta }_1(⌜),\hfill \\ \hfill {\zeta }_2\left((-\overline{\mathcal{F}})(⌜)\right)& ={\zeta }_2(-\overline{\mathcal{F}}(⌜))={\zeta }_2\left(\overline{\mathcal{F}}(⌜)\right)\le {\zeta }_1(⌜).\hfill \end{array}$$

Therefore, the neutral map $\overline{0}$ and additive inverses exist, satisfying

$$\overline{\mathcal{F}}+\overline{0}=\overline{0}\oplus \overline{\mathcal{F}}=\overline{\mathcal{F}},\overline{\mathcal{F}}+(-\overline{\mathcal{F}})=(-\overline{\mathcal{F}})+\overline{\mathcal{F}}=\overline{0}.$$

Hence, $\mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ forms an abelian group under addition. We now define scalar multiplication: for $\overline{\mathcal{F}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ and $⌞\in \mathfrak{R}$, set

$$(⌞·\overline{\mathcal{F}})\left(a\right):=\overline{\mathcal{F}}(⌞⌜),\forall ⌜\in M.$$

This assignment is a valid morphism because the function $⌜\mapsto \overline{\mathcal{F}}(⌞⌜)$ respects the module structure, and

$$\begin{array}{cc}\hfill {\alpha }_2\left((⌞\overline{\mathcal{F}})(⌜)\right)& ={\alpha }_2\left(\overline{\mathcal{F}}(⌞⌜)\right)\ge {\alpha }_1(⌞⌜)\ge {\alpha }_1(⌜),\hfill \\ \hfill {\beta }_2\left((⌞\overline{\mathcal{F}})(⌜)\right)& ={\beta }_2\left(\overline{\mathcal{F}}(⌞⌜)\right)\le {\beta }_1(⌞⌜)\le {\beta }_1(⌜),\hfill \\ \hfill {\zeta }_2\left((⌞\overline{\mathcal{F}})(⌜)\right)& ={\zeta }_2\left(\overline{\mathcal{F}}(⌞⌜)\right)\le {\zeta }_1(⌞⌜)\le {\zeta }_1(⌜).\hfill \end{array}$$

Thus, $⌞\overline{\mathcal{F}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$. If $\mathfrak{R}$ is a commutative ring, the following module properties are satisfied for all $⌞,{⌞}_1,{⌞}_2\in \mathfrak{R}$:

$$\begin{array}{cc}\hfill \left(⌞(\overline{\mathcal{F}}+\overline{\mathcal{H}})\right)& =⌞\overline{\mathcal{F}}+⌞\overline{\mathcal{H}},\hfill \\ \hfill ({⌞}_1+{⌞}_2)\overline{\mathcal{F}}& ={⌞}_1\overline{\mathcal{F}}+{⌞}_2\overline{\mathcal{F}},\hfill \\ \hfill \left({⌞}_1{⌞}_2\right)\overline{\mathcal{F}}& ={⌞}_1\left({⌞}_2\overline{\mathcal{F}}\right),\hfill \\ \hfill 1·\overline{\mathcal{F}}& =\overline{\mathcal{F}}.\hfill \end{array}$$

Therefore, under these operations, $\mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ is a left R-module when $\mathfrak{R}$ is commutative. □

Next, we consider ${\mathcal{P}}_M$, ${\mathcal{S}}_N$ and ${\mathcal{J}}_J=\{<⌝,{\alpha }_3(⌝),{\beta }_3(⌝),{\zeta }_3(⌝)>\mid ⌝\in J\}$ represent an arbitrary neutrosophic left (resp., right) $\mathfrak{R}$-module, where J denotes a left (or right) $\mathfrak{R}$-module; also, we define

$${\mathcal{P}}_M\stackrel{\overline{\mathcal{F}}}{\to }{\mathcal{S}}_N\stackrel{\overline{\mathcal{H}}}{\to }{\mathcal{J}}_J,$$

Since

$$\begin{array}{cc}\hfill {\alpha }_3\left(\left(\mathcal{H}\mathcal{F}\right)(⌜)\right)& ={\alpha }_3(\mathcal{H}\left(\mathcal{F}(⌜)\right)\ge {\alpha }_1\left(\mathcal{F}(⌜)\right)\ge {\alpha }_2(⌜),\hfill \\ \hfill {\beta }_3\left(\left(\mathcal{H}\mathcal{F}\right)(⌜)\right)& ={\beta }_3(\mathcal{H}\left(\mathcal{F}(⌜)\right)\le {\beta }_1\left(\mathcal{F}(⌜)\right)\le {\beta }_2(⌜),\hfill \\ \hfill {\zeta }_3\left(\left(\mathcal{H}\mathcal{F}\right)(⌜)\right)& ={\zeta }_3(\mathcal{H}\left(\mathcal{F}(⌜)\right)\le {\zeta }_1\left(\mathcal{F}(⌜)\right)\le {\zeta }_2(⌜),\hfill \end{array}$$

where $⌜\in M.$

Now, we can present the definition of composition as follows

$$\overline{\mathcal{F}}\circ \overline{\mathcal{H}}=\overline{\mathcal{F}}\overline{\mathcal{H}}.$$

Next, we constitute the neutrosophic category.

Definition 7.

We define the category ${\mathcal{C}}_m$, referred to as the category of neutrosophic $\mathfrak{R}$-modules, in the following manner:

1. 

The category ${\mathcal{C}}_m$ is formed by a collection of objects

$$\mathrm{Obj}\left({\mathcal{C}}_m\right)=\{{\mathcal{P}}_M,{\mathcal{S}}_N,\dots \},$$

where each element represents a complete neutrosophic $\mathfrak{R}$-module.

2. 

For every ordered pair of objects $({\mathcal{P}}_M,{\mathcal{S}}_N)$, we associate a set of morphisms

$$\mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N),$$

with composition maps

$$\mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)\times \mathrm{Hom}({\mathcal{S}}_N,{\mathcal{J}}_J)⟶\mathrm{Hom}({\mathcal{P}}_M,{\mathcal{J}}_J),$$

with $(\overline{\mathcal{F}},\overline{\mathcal{H}})\to \overline{\mathcal{F}\mathcal{H}}$, where $\overline{\mathcal{F}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ and $\overline{\mathcal{H}}\in \mathrm{Hom}({\mathcal{S}}_N,{\mathcal{J}}_J)$.

These morphisms satisfy the following axioms:

(a) 

For each object ${\mathcal{P}}_M\in \mathrm{Obj}\left({\mathcal{C}}_m\right)$, there exists an identity morphism

$$\overline{{\mathrm{id}}_{{\mathcal{P}}_M}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{P}}_M).$$

(b) 

The composition of morphisms is associative.

Subsequently, we examine an idea related to the morphisms within the category ${\mathcal{C}}_m$.

Definition 8.

Let $\overline{\mathcal{F}}$$\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ be a neutrosophic $\mathfrak{R}$-map. We say that $\overline{\mathcal{F}}$ is neutrosophic split if there exists a morphism $\overline{\mathcal{H}}$$\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ such that the composition $\overline{\mathcal{F}}\circ \overline{\mathcal{H}}=\overline{\mathcal{F}}\overline{\mathcal{H}}=\overline{{\mathrm{id}}_{{\mathcal{P}}_M}}$.

Theorem 1.

Let $\overline{\mathcal{F}}$$\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$. The mapping $\overline{\mathcal{F}}$ is a neutrosophic split morphism if and only if

$$\begin{array}{cc}\hfill {\alpha }_2\left(\theta \right)& =max\left\{{\alpha }_1\left(\sigma \right)\right|\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},\hfill \\ \hfill {\beta }_2\left(\theta \right)& =min\left\{{\beta }_1\left(\sigma \right)\right|\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},\hfill \\ \hfill {\zeta }_2\left(\theta \right)& =min\left\{{\zeta }_1\left(\sigma \right)\right|\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},\hfill \end{array}$$

where $\theta \in N$ and ${\mathcal{F}}^{-1}\left(\theta \right)$ denotes the complete set of elements whose image under $\mathcal{F}$ equals θ.

Proof. 

By the definition of $\overline{\mathcal{F}}$ as a neutrosophic split morphism, there exists a morphism $\overline{\mathcal{H}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ such that $\overline{\mathcal{F}}\overline{\mathcal{H}}=\overline{{\mathrm{id}}_{{\mathcal{P}}_M}}$. For an arbitrary $\theta \in N$, we have

$$\begin{array}{c}\hfill {\alpha }_2\left(\theta \right)={\alpha }_2\left(\left(\mathcal{F}\mathcal{H}\right)\left(\theta \right)\right)\ge {\alpha }_1\left(\mathcal{H}\left(\theta \right)\right)\ge {\alpha }_2\left(\theta \right),\\ \hfill {\beta }_2\left(\theta \right)={\beta }_2\left(\left(\mathcal{F}\mathcal{H}\right)\left(\theta \right)\right)\le {\beta }_1\left(\mathcal{H}\left(\theta \right)\right)\le {\beta }_2\left(\theta \right),\\ \hfill {\zeta }_2\left(\theta \right)={\zeta }_2\left(\left(\mathcal{F}\mathcal{H}\right)\left(\theta \right)\right)\le {\zeta }_1\left(\mathcal{H}\left(\theta \right)\right)\le {\zeta }_2\left(\theta \right).\end{array}$$

Hence, ${\alpha }_1\left(\mathcal{H}\left(\theta \right)\right)={\alpha }_2\left(\theta \right)$, ${\beta }_1\left(\mathcal{H}\left(\theta \right)\right)={\beta }_2\left(\theta \right)$ and ${\zeta }_1\left(\mathcal{H}\left(\theta \right)\right)={\zeta }_2\left(\theta \right)$. Since $\mathcal{F}\left(\mathcal{H}\right(\theta \left)\right)=\theta$, it follows that $\mathcal{H}\left(\theta \right)\in {\mathcal{F}}^{-1}\left(\theta \right)$. Therefore, ${\alpha }_2\left(\theta \right)={\alpha }_1\left(\sigma \right)$, ${\beta }_2\left(\theta \right)={\beta }_1\left(\sigma \right)$ and ${\zeta }_2\left(\theta \right)={\zeta }_1\left(\sigma \right)$ for $\sigma =\mathcal{H}\left(\theta \right)\in {\mathcal{F}}^{-1}\left(\theta \right)$.

Conversely, if $\mathcal{F}\left({\sigma }^{\prime }\right)=\theta$ for some ${\sigma }^{\prime }\in {\mathcal{F}}^{-1}\left(\theta \right)$, then

$$\begin{array}{cc}\hfill {\alpha }_2\left(\theta \right)& ={\alpha }_2\left(\mathcal{F}\left({\sigma }^{\prime }\right)\right)\ge {\alpha }_1\left({\sigma }^{\prime }\right),\hfill \\ \hfill {\beta }_2\left(\theta \right)& ={\beta }_2\left(\mathcal{F}\left({\sigma }^{\prime }\right)\right)\le {\beta }_1\left({\sigma }^{\prime }\right),\hfill \\ \hfill {\zeta }_2\left(\theta \right)& ={\zeta }_2\left(\mathcal{F}\left({\sigma }^{\prime }\right)\right)\le {\zeta }_1\left({\sigma }^{\prime }\right),\hfill \end{array}$$

From this, it follows that

$$\begin{array}{cc}\hfill {\alpha }_2& =max\left\{{\alpha }_1\left(\sigma \right)\right|\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},\hfill \\ \hfill {\beta }_2& =min\left\{{\beta }_1\left(\sigma \right)\right|\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},\hfill \\ \hfill {\zeta }_2& =min\left\{{\zeta }_1\left(\sigma \right)\right|\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\}.\hfill \end{array}$$

Now, we proceed to prove the sufficiency part. Let us construct an $\mathfrak{R}$-morphism $\mathcal{H}:N⟶M$ by assigning to each $\theta \in N$ an element $\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)$ such that

$${\alpha }_1\left(\sigma \right)\ge {\alpha }_1\left({\sigma }^{\prime }\right),{\beta }_1\left(\sigma \right)\le {\beta }_1\left({\sigma }^{\prime }\right),\mathrm{and}{\zeta }_1\left(\sigma \right)\le {\zeta }_1\left({\sigma }^{\prime }\right)\forall {\sigma }^{\prime }\in {\mathcal{F}}^{-1}\left(\theta \right).$$

For any $\sigma \in N$, we have $\mathcal{F}\mathcal{H}\left(\theta \right)=\mathcal{F}\left(\sigma \right)=\theta$, which implies $\mathcal{F}\mathcal{H}={\mathrm{id}}_{{\mathcal{P}}_M}$. Furthermore,

$$\begin{array}{cc}\hfill {\alpha }_1\left(\mathcal{H}\left(\theta \right)\right)& ={\alpha }_1\left(\sigma \right)=max\left\{{\alpha }_1\left({\sigma }^{\prime }\right)\right|{\sigma }^{\prime }\in {\mathcal{F}}^{-1}\left(\theta \right)\}={\alpha }_2\left(\theta \right),\hfill \\ \hfill {\beta }_1\left(\mathcal{H}\left(\theta \right)\right)& ={\beta }_1\left(\sigma \right)=min\left\{{\beta }_1\left({\sigma }^{\prime }\right)\right|{\sigma }^{\prime }\in {\mathcal{F}}^{-1}\left(\theta \right)\}={\beta }_2\left(\theta \right),\hfill \\ \hfill {\zeta }_1\left(\mathcal{H}\left(\theta \right)\right)& ={\zeta }_1\left(\sigma \right)=min\left\{{\zeta }_1\left({\sigma }^{\prime }\right)\right|{\sigma }^{\prime }\in {\mathcal{F}}^{-1}\left(\theta \right)\}={\zeta }_2\left(\theta \right).\hfill \end{array}$$

Therefore, the requirement stated in condition (2) of Definition 6 holds true. Consequently, there exists a neutrosophic $\mathfrak{R}$-morphism $\overline{\mathcal{H}}\in \mathrm{Hom}({\mathcal{P}}_M,{\mathcal{S}}_N)$ such that

$$\overline{\mathcal{F}}\overline{\mathcal{H}}=\overline{{\mathrm{id}}_{{\mathcal{P}}_M}}.$$

Thus, $\overline{\mathcal{F}}$ is indeed a neutrosophic split morphism. □

Remark 1.

In Theorem 1, the expressions involving ${\mathcal{F}}^{-1}\left(\theta \right)$ are defined through the supremum and infimum of the corresponding neutrosophic membership functions over the entire preimage of θ. That is, the values

$${\alpha }_2\left(\theta \right)=max\{{\alpha }_1\left(\sigma \right)\mid \sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},$$

$${\beta }_2\left(\theta \right)=min\{{\beta }_1\left(\sigma \right)\mid \sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},$$

$${\zeta }_2\left(\theta \right)=min\{{\zeta }_1\left(\sigma \right)\mid \sigma \in {\mathcal{F}}^{-1}\left(\theta \right)\},$$

do not depend on any particular representative $\sigma \in {\mathcal{F}}^{-1}\left(\theta \right)$, but on the complete set of preimages. Hence, the induced map $\overline{\mathcal{F}}:{\mathcal{S}}_N\to {\mathcal{P}}_M$ is well-defined and preserves $\mathfrak{R}$-linearity.

3.2. Neutrosophic Finitely Generated Modules

In what follows, we introduce a collection of fundamental definitions concerning singular and fully neutrosophic $\mathfrak{R}$-modules, which will serve as a basis for the subsequent discussion.

Definition 9.

A neutrosophic $\mathfrak{R}$-module is called a singular neutrosophic $\mathfrak{R}$-module if

$$\alpha \left(\theta \right)=1,\beta \left(\theta \right)=0and\zeta \left(\theta \right)=0forall\theta \in M.$$

Proposition 1.

Let M be a finitely generated $\mathfrak{R}$-module. ${\mathcal{P}}_M$ is singular if and only if there exists a generating set $\{{\mathfrak{m}}_i\in M\mid i=1,2,\dots ,n\}$ satisfying

$$\alpha \left({\mathfrak{m}}_i\right)=1,\beta \left({\mathfrak{m}}_i\right)=0and\zeta \left({\mathfrak{m}}_i\right)=0foralli=1,2,\dots ,n.$$

Proof. 

Suppose that $\{{\mathfrak{m}}_1,{\mathfrak{m}}_2,\dots ,{\mathfrak{m}}_n\}$ is a generating set of M. Take any $\theta \in M$. Then, $\theta$ can be expressed as

$$\theta =\sum _{i=1}^n{c}_i{\mathfrak{m}}_i,\mathrm{with}{c}_i\in \mathfrak{R}.$$

Now, we have

$$\begin{array}{cc}\hfill {\alpha }_1\left(\theta \right)={\alpha }_1\left(\sum _{i=1}^n{c}_i{\mathfrak{m}}_i\right)& \ge min\{{\alpha }_1\left(c_i{\mathfrak{m}}_i\right)\mid i=1,2,\dots ,n\}\hfill \\ & \ge min\{{\alpha }_1\left({\mathfrak{m}}_i\right)\mid i=1,2,\dots ,n\}=1,\hfill \\ \hfill {\beta }_1\left(\theta \right)={\beta }_1\left(\sum _{i=1}^n{c}_i{\mathfrak{m}}_i\right)& \le max\{{\beta }_1\left(c_i{\mathfrak{m}}_i\right)\mid i=1,2,\dots ,n\}\hfill \\ & \le max\{{\beta }_1\left({\mathfrak{m}}_i\right)i=1,2,\dots ,n\}=0,\hfill \\ \hfill {\zeta }_1\left(\theta \right)={\zeta }_1\left(\sum _{i=1}^n{c}_i{\mathfrak{m}}_i\right)& \le max\{{\zeta }_1\left(c_i{\mathfrak{m}}_i\right)\mid i=1,2,\dots ,n\}\hfill \\ & \le max\{{\zeta }_1\left({\mathfrak{m}}_i\right)\mid i=1,2,\dots ,n\}=0.\hfill \end{array}$$

Thus, $\alpha \left(\theta \right)=1$, $\beta \left(\theta \right)=1$ and $\zeta \left(\theta \right)=1$ for all $\theta \in M$, which implies ${\mathcal{P}}_M$ is singular. □

Definition 10.

A neutrosophic $\mathfrak{R}$-module is called a fully neutrosophic $\mathfrak{R}$-module if

$$\alpha \left(\theta \right)=0,\beta \left(\theta \right)=1and\zeta \left(\theta \right)=1forall0\ne \theta \in M.$$

Example 1.

Let $\mathfrak{R}=\mathfrak{Z}$ and $M=\mathfrak{Z}$ and define a neutrosophic $\mathfrak{Z}$-module ${\mathcal{M}}_{\mathfrak{Z}}=\{<\iota ,{\mu }_{\mathfrak{Z}}\left(\iota \right),{\gamma }_{\mathfrak{Z}}\left(\iota \right),{\zeta }_{\mathfrak{Z}}\left(\iota \right)>:\iota \in \mathfrak{Z}\}$ as follows:

$${\mu }_{\mathfrak{Z}}\left(\iota \right)=\left\{\begin{array}{cc}1\hfill & if\iota =0,\hfill \\ 0.6\hfill & otherwise,\hfill \end{array}\right.,{\gamma }_{\mathfrak{Z}}\left(\iota \right)=\left\{\begin{array}{cc}0\hfill & if\iota =0,\hfill \\ 0.7\hfill & otherwise,\hfill \end{array}\right.,{\zeta }_{\mathfrak{Z}}\left(\iota \right)=\left\{\begin{array}{cc}0\hfill & if\iota =0,\hfill \\ 0.3\hfill & otherwise,\hfill \end{array}\right.$$

where $\mathfrak{Z}$ is an integral number. ${\mathcal{M}}_{\mathfrak{Z}}$ is a finitely generated neutrosophic $\mathfrak{Z}$-module that is neither a singular nor fully neutrosophic $\mathfrak{Z}$-module.

Theorem 2.

Let ${\mathcal{P}}_M$ be a neutrosophic $\mathfrak{R}$-module that is finitely generated. Then there exists a generating set $\{{\mathfrak{m}}_i\mid i=1,2,\dots ,n\}$ such that

$$\begin{array}{cc}\hfill {\alpha }_1\left({\mathfrak{m}}_1\right)& ={\alpha }_1\left({\mathfrak{m}}_2\right)=\dots ={\alpha }_1\left({\mathfrak{m}}_n\right)=min\{{\alpha }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\beta }_1\left({\mathfrak{m}}_1\right)& ={\beta }_1\left({\mathfrak{m}}_2\right)=\dots ={\beta }_1\left({\mathfrak{m}}_n\right)=max\{{\beta }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\zeta }_1\left({\mathfrak{m}}_1\right)& ={\zeta }_1\left({\mathfrak{m}}_2\right)=\dots ={\zeta }_1\left({\mathfrak{m}}_n\right)=max\{{\zeta }_1\left(\theta \right)\mid \theta \in M\}.\hfill \end{array}$$

Proof. 

Select any generating set $\{{\mathfrak{b}}_i\mid i=1,2,\dots ,n\}$ of M. Let $\theta \in M$ be written as

$$\theta =\sum _{i=1}^n{\mathfrak{r}}_i{\mathfrak{b}}_i,{\mathfrak{r}}_i\in \mathfrak{R}(i=1,2,\dots ,n).$$

By the properties of ${\mathcal{P}}_M$, we obtain

$$\begin{array}{cc}\hfill {\alpha }_1\left(\theta \right)& ={\alpha }_1\left(\sum _{i=1}^n{\mathfrak{r}}_i{\mathfrak{b}}_i\right)\ge min\{{\alpha }_1\left({\mathfrak{r}}_i{\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}\ge min\{{\alpha }_1\left({\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\},\hfill \\ \hfill {\beta }_1\left(\theta \right)& ={\beta }_1\left(\sum _{i=1}^n{\mathfrak{r}}_i{\mathfrak{b}}_i\right)\le max\{{\beta }_1\left({\mathfrak{r}}_i{\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}\le max\{{\beta }_1\left({\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\},\hfill \\ \hfill {\zeta }_1\left(\theta \right)& ={\zeta }_1\left(\sum _{i=1}^n{\mathfrak{r}}_i{\mathfrak{b}}_i\right)\le max\{{\zeta }_1\left({\mathfrak{r}}_i{\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}\le max\{{\zeta }_1\left({\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}.\hfill \end{array}$$

Without loss of generality, suppose that $min\{{\alpha }_1\left({\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}={\alpha }_1\left({\mathfrak{b}}_1\right)$, $max\{{\beta }_1\left({\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}={\beta }_1\left({\mathfrak{b}}_1\right)$, and $max\{{\zeta }_1\left({\mathfrak{b}}_i\right)\mid i=1,2,\dots ,n\}={\zeta }_1\left({\mathfrak{b}}_1\right)$, that is,

$$\begin{array}{cc}\hfill {\alpha }_1\left({\mathfrak{b}}_1\right)& =min\{{\alpha }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\beta }_1\left({\mathfrak{b}}_1\right)& =max\{{\beta }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\zeta }_1\left({\mathfrak{b}}_1\right)& =max\{{\zeta }_1\left(\theta \right)\mid \theta \in M\}.\hfill \end{array}$$

Set ${\mathfrak{m}}_1={\mathfrak{b}}_1$. If there exists some ${\mathfrak{b}}_2$ with either ${\alpha }_1\left({\mathfrak{b}}_2\right)\ne {\alpha }_1\left({\mathfrak{b}}_1\right)$, ${\beta }_1\left({\mathfrak{b}}_2\right)\ne {\beta }_1\left({\mathfrak{b}}_1\right)$, or ${\zeta }_1\left({\mathfrak{b}}_2\right)\ne {\zeta }_1\left({\mathfrak{b}}_1\right)$, select such a ${\mathfrak{b}}_2$. If ${\alpha }_1\left({\mathfrak{b}}_2\right)>{\alpha }_1\left({\mathfrak{b}}_1\right)$, ${\beta }_1\left({\mathfrak{b}}_2\right)<{\beta }_1\left({\mathfrak{b}}_1\right)$, and ${\zeta }_1\left({\mathfrak{b}}_2\right)<{\zeta }_1\left({\mathfrak{b}}_1\right)$ proceed as follows: For any $\theta \in M$ written as

$$\theta =({\mathfrak{r}}_1-{\mathfrak{r}}_2){\mathfrak{b}}_1+{\mathfrak{r}}_2({\mathfrak{b}}_1+{\mathfrak{b}}_2)+{\mathfrak{r}}_3{\mathfrak{b}}_3+\dots +{\mathfrak{r}}_n{\mathfrak{b}}_n,$$

the set $\{{\mathfrak{b}}_1,{\mathfrak{b}}_1+{\mathfrak{b}}_2,{\mathfrak{b}}_3,\dots ,{\mathfrak{b}}_n\}$ also generates M.

Now, note that

$$\begin{array}{cc}\hfill {\alpha }_1\left({\mathfrak{b}}_1\right)& ={\alpha }_1(({\mathfrak{b}}_1+{\mathfrak{b}}_2)+(-{\mathfrak{b}}_2))\ge min\{{\alpha }_1({\mathfrak{b}}_1+{\mathfrak{b}}_2),{\alpha }_1(-{\mathfrak{b}}_2)\}\ge min\{{\alpha }_1({\mathfrak{b}}_1+{\mathfrak{b}}_2),{\alpha }_1\left({\mathfrak{b}}_2\right)\},\hfill \\ \hfill {\beta }_1\left({\mathfrak{b}}_1\right)& ={\beta }_1(({\mathfrak{b}}_1+{\mathfrak{b}}_2)+(-{\mathfrak{b}}_2))\le max\{{\beta }_1({\mathfrak{b}}_1+{\mathfrak{b}}_2),{\beta }_1(-{\mathfrak{b}}_2)\}\le max\{{\beta }_1({\mathfrak{b}}_1+{\mathfrak{b}}_2),{\beta }_1\left({\mathfrak{b}}_2\right)\},\hfill \\ \hfill {\zeta }_1\left({\mathfrak{b}}_1\right)& ={\zeta }_1(({\mathfrak{b}}_1+{\mathfrak{b}}_2)+(-{\mathfrak{b}}_2))\le max\{{\zeta }_1({\mathfrak{b}}_1+{\mathfrak{b}}_2),{\zeta }_1(-{\mathfrak{b}}_2)\}\le max\{{\zeta }_1({\mathfrak{b}}_1+{\mathfrak{b}}_2),{\zeta }_1\left({\mathfrak{b}}_2\right)\}.\hfill \end{array}$$

Since ${\alpha }_1\left({\mathfrak{b}}_2\right)>{\alpha }_1\left({\mathfrak{b}}_1\right)$, ${\beta }_1\left({\mathfrak{b}}_2\right)<{\beta }_1\left({\mathfrak{b}}_1\right)$, and ${\zeta }_1\left({\mathfrak{b}}_2\right)<{\zeta }_1\left({\mathfrak{b}}_1\right)$, it follows that ${\alpha }_1\left({\mathfrak{b}}_1\right)={\alpha }_1({\mathfrak{b}}_1+{\mathfrak{b}}_1)$, ${\beta }_1\left({\mathfrak{b}}_1\right)={\beta }_1({\mathfrak{b}}_1+{\mathfrak{b}}_1)$ and ${\zeta }_1\left({\mathfrak{b}}_1\right)={\zeta }_1({\mathfrak{b}}_1+{\mathfrak{b}}_1)$. In this case, define ${\mathfrak{m}}_2={\mathfrak{b}}_1+{\mathfrak{b}}_2$, and observe that

$$\begin{array}{cc}\hfill {\alpha }_1\left({\mathfrak{b}}_1\right)& ={\alpha }_1\left({\mathfrak{m}}_2\right)=min\{{\alpha }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\beta }_1\left({\mathfrak{b}}_1\right)& ={\beta }_1\left({\mathfrak{m}}_2\right)=max\{{\beta }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\zeta }_1\left({\mathfrak{b}}_1\right)& ={\zeta }_1\left({\mathfrak{m}}_2\right)=max\{{\zeta }_1\left(\theta \right)\mid \theta \in M\}.\hfill \end{array}$$

Continuing this process, we eventually obtain a generating set

$$\{{\mathfrak{m}}_1,{\mathfrak{m}}_2,\dots ,{\mathfrak{m}}_n\},$$

satisfying

$$\begin{array}{cc}\hfill {\alpha }_1\left({\mathfrak{m}}_1\right)& ={\alpha }_1\left({\mathfrak{m}}_2\right)=\dots ={\alpha }_1\left({\mathfrak{m}}_n\right)=min\{{\alpha }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\beta }_1\left({\mathfrak{m}}_1\right)& ={\beta }_1\left({\mathfrak{m}}_2\right)=\dots ={\beta }_1\left({\mathfrak{m}}_n\right)=max\{{\beta }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\zeta }_1\left({\mathfrak{m}}_1\right)& ={\zeta }_1\left({\mathfrak{m}}_2\right)=\dots ={\zeta }_1\left({\mathfrak{m}}_n\right)=max\{{\zeta }_1\left(\theta \right)\mid \theta \in M\}.\hfill \end{array}$$

□

Remark 2.

This specific generating set is known as the set of neutrosophic homogeneous generators of M.

It is established that an $\mathfrak{R}$-module M is free if and only if

$$M\cong \bigcup _{i\in I}\mathfrak{R}{\mathfrak{x}}_i,$$

where $\mathfrak{R}{\mathfrak{x}}_i\cong \mathfrak{R}$ and $\{{\mathfrak{x}}_i\mid i\in I\}$ is a basis for M. From Theorem 2 and the definition of a free $\mathfrak{R}$-module, we deduce the following.

Corollary 1.

Let ${\mathcal{P}}_M$ be a neutrosophic finitely generated free $\mathfrak{R}$-module. Then there exists a neutrosophic homogeneous basis

$$\{{\mathfrak{x}}_i\mid i=1,2,\dots ,n\},$$

such that

$$\begin{array}{cc}\hfill {\alpha }_1\left({\mathfrak{x}}_1\right)& ={\alpha }_1\left({\mathfrak{x}}_2\right)=\dots ={\alpha }_1\left({\mathfrak{x}}_n\right)=min\{{\alpha }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\beta }_1\left({\mathfrak{x}}_1\right)& ={\beta }_1\left({\mathfrak{x}}_2\right)=\dots ={\beta }_1\left({\mathfrak{x}}_n\right)=max\{{\beta }_1\left(\theta \right)\mid \theta \in M\},\hfill \\ \hfill {\zeta }_1\left({\mathfrak{x}}_1\right)& ={\zeta }_1\left({\mathfrak{x}}_2\right)=\dots ={\zeta }_1\left({\mathfrak{x}}_n\right)=max\{{\zeta }_1\left(\theta \right)\mid \theta \in M\}.\hfill \end{array}$$

3.3. Neutrosophic Modules with Finite Neutrosophic Value

Definition 11.

For $\tau \in [0,1]$, define ${\mathcal{M}}_{M\tau }=\{<\theta ,{\alpha }_{\tau }\left(\theta \right),{\beta }_{\tau }\left(\theta \right),{\zeta }_{\tau }\left(\theta \right)>\mid \theta \in M\}$ such that

$$\begin{array}{cc}\hfill {\alpha }_{M_{\tau }}& =\{\theta \in M\mid \alpha (\theta )\ge \tau \},\hfill \\ \hfill {\beta }_{M_{\tau }}& =\{\theta \in M\mid \beta (\theta )\le \tau \},\hfill \\ \hfill {\zeta }_{M_{\tau }}& =\{\theta \in M\mid \zeta (\theta )\le \tau \}.\hfill \end{array}$$

This corresponds to the τ-level cut used in neutrosophic set theory. Also, ${\mathcal{M}}_{M\tau }$ is itself a neutrosophic sub-module over M.

Clearly, ${\mathcal{M}}_{M_0}=\{\theta \in M\mid <{\alpha }_0\left(\theta \right),{\beta }_0\left(\theta \right),{\zeta }_0\left(\theta \right)>=<\alpha \left(\theta \right),0,0>\}$ and ${\mathcal{M}}_{M_1}=\{\theta \in M\mid <{\alpha }_1\left(\theta \right),{\beta }_1\left(\theta \right),{\zeta }_1\left(\theta \right)>=<1,\beta \left(\theta \right),\zeta \left(\theta \right)>\}$.

Lemma 2.

Let $\mathfrak{R}$ be a principal ideal domain and let ${\mathcal{M}}_M$ be a neutrosophic sub-module of a finitely generated free $\mathfrak{R}$-module M. Then there exists a finite sequence of neutrosophic sub-modules beginning with ${\mathcal{M}}_{M_1}$ and ending with ${\mathcal{M}}_{M_0}$. Moreover, the maximal chain in such sequences is unique.

Proof. 

We start by noting that if $0<{\tau }_1<{\tau }_2\le 1$, then

$$\begin{array}{cc}\hfill {\alpha }_{M{\tau }_1}& \ge {\alpha }_{M{\tau }_2},\hfill \\ \hfill {\beta }_{M{\tau }_1}& \le {\beta }_{M{\tau }_2},\hfill \\ \hfill {\zeta }_{M{\tau }_1}& \le {\zeta }_{M{\tau }_2},\hfill \end{array}$$

Choose any $\tau \in (0,1)$. If $M_{\tau }=M_0$ or $M_{\tau }=M_{M_1}$; we choose another $\tau$ and proceed.

• Case 1: Assume $M_{\tau }=M_1$ for all $\tau \in (0,1)$. Then $M_{\tau }=M_0=M_1$. Take some $⌜\in M$ with $\alpha (⌜)<1$, $\beta (⌜)>1$ and $\zeta (⌜)>1$. Let ${\tau }_2={\displaystyle \frac{{\tau }_1+1}{2}}$, which lies in $(0,1)$ and satisfies ${\tau }_1<{\tau }_2$. Since $⌜\in M_{{\tau }_1}$ but $⌜\notin M_{{\tau }_2}$, we contradict $M_{{\tau }_1}=M_{{\tau }_2}$. Therefore, this case yields $M_{\tau }=M_0=M_1$ and the chain has only one element.
• Case 2: Suppose $M_{\tau }=M_0$ for all $\tau \in (0,1)$. If $\{⌜\in M\mid \alpha (⌜)=\beta (⌜)=\zeta (⌜)=0\}\ne ⌀$, then this is analogous to Case 1. Otherwise, there is a single maximal chain

  $$\begin{array}{cc}\hfill {\alpha }_{M_1}& \le {\alpha }_{M_0}={\alpha }_M,\hfill \\ \hfill {\beta }_{M_1}& \ge {\beta }_{M_0}={\beta }_M,\hfill \\ \hfill {\zeta }_{M_1}& \ge {\zeta }_{M_0}={\zeta }_M.\hfill \end{array}$$

In the general situation, there exists $\tau \in (0,1)$ such that

$$\begin{array}{cc}\hfill {\alpha }_{M_1}& \le {\alpha }_{M_{\tau }}\le {\alpha }_{M_0}={\alpha }_M,\hfill \\ \hfill {\beta }_{M_1}& \ge {\beta }_{M_{\tau }}\ge {\beta }_{M_0}={\beta }_M,\hfill \\ \hfill {\zeta }_{M_1}& \ge {\zeta }_{M_{\tau }}\ge {\zeta }_{M_0}={\zeta }_M.\hfill \end{array}$$

Now, by iterating this reasoning over sub-intervals of $(0,1)$, one can obtain a sequence of parameters

$$0<{\tau }_1<{\tau }_2<\dots <{\tau }_p<1,$$

with

$$\begin{array}{cc}\hfill {\alpha }_{M_1}& \le {\alpha }_{M_{{\tau }_p}}\le \dots \le {\alpha }_{M_{{\tau }_2}}\le {\alpha }_{M_{{\tau }_1}}\le {\alpha }_{M_0},\hfill \\ \hfill {\beta }_{M_1}& \ge {\beta }_{M_{{\tau }_p}}\ge \dots \ge {\beta }_{M_{{\tau }_2}}\ge {\beta }_{M_{{\tau }_1}}\ge {\beta }_{M_0},\hfill \\ \hfill {\zeta }_{M_1}& \ge {\zeta }_{M_{{\tau }_p}}\ge \dots \ge {\zeta }_{M_{{\tau }_2}}\ge {\zeta }_{M_{{\tau }_1}}\ge {\zeta }_{M_0},\hfill \end{array}$$

(1)

The number of steps p in (1) must satisfy $p\le n-1$, where $n=rank\left(M\right)$.

If $p\ge n$ or $p=\infty$, then since $\mathfrak{R}$ is a principal ideal domain, the strict inclusion ${\mathcal{M}}_{M_{{\tau }_i}}⊊{\mathcal{M}}_{M_{{\tau }_{i-1}}}$ would imply that

$$rank\left(M_{{\tau }_i}\right)<rank\left(M_{{\tau }_{i-1}}\right).$$

Because M has finite rank n, this would yield the decreasing chain

$$0\le rank\left(M_{{\tau }_p}\right)<rank\left(M_{{\tau }_{p-1}}\right)<\dots <rank\left(M_{{\tau }_1}\right)<rank\left(M\right)=n,$$

which cannot extend beyond n steps.

Consequently, any maximal chain must stop within $n-1$ strict inclusions. Such a chain is characterized by the following: for every $\tau \in (0,1)$, there exists some index $j\in \{1,2,\dots ,p\}$ such that

$$\begin{array}{cc}\hfill {\alpha }_{M_{\tau }}& ={\alpha }_{M_{{\tau }_j}},\hfill \\ \hfill {\beta }_{M_{\tau }}& ={\beta }_{M_{{\tau }_j}},\hfill \\ \hfill {\zeta }_{M_{\tau }}& ={\zeta }_{M_{{\tau }_j}}.\hfill \end{array}$$

Thus, only one maximal chain arises, and the chain among neutrosophic sub-modules with ${\mathcal{M}}_M$ at the top is uniquely determined as follows:

$$\begin{array}{cc}\hfill {\alpha }_{M_1}& \le {\alpha }_{M_{{\Gamma }_p}}\le \dots \le {\alpha }_{M_{{\Gamma }_2}}\le {\alpha }_{M_{{\Gamma }_1}}\le {\alpha }_{M_0},\hfill \\ \hfill {\beta }_{M_1}& \ge {\beta }_{M_{{\Gamma }_p}}\ge \dots \ge {\beta }_{M_{{\Gamma }_2}}\ge {\beta }_{M_{{\Gamma }_1}}\ge {\beta }_{M_0},\hfill \\ \hfill {\zeta }_{M_1}& \ge {\zeta }_{M_{{\Gamma }_p}}\ge \dots \ge {\zeta }_{M_{{\Gamma }_2}}\ge {\zeta }_{M_{{\Gamma }_1}}\ge {\zeta }_{M_0}.\hfill \end{array}$$

(2)

Also, we get ${\alpha }_{M_{{\Gamma }_i}}={\alpha }_{M_{{\tau }_j}}$, ${\beta }_{M_{{\Gamma }_i}}={\beta }_{M_{{\tau }_j}}$ and ${\zeta }_{M_{{\Gamma }_i}}={\zeta }_{M_{{\tau }_j}}$ when ${\Gamma }_i={\tau }_j$. Continuing from the chain (1), we observe that

$$\begin{array}{cc}\hfill {\alpha }_{M_{{\Gamma }_i}}& ={\alpha }_{M_{{\tau }_j}}\mathrm{or}{\alpha }_{M_{{\Gamma }_i}}={\alpha }_{M_{{\tau }_{j+1}}},\hfill \\ \hfill {\beta }_{M_{{\Gamma }_i}}& ={\beta }_{M_{{\tau }_j}}\mathrm{or}{\beta }_{M_{{\Gamma }_i}}={\beta }_{M_{{\tau }_{j+1}}},\hfill \\ \hfill {\zeta }_{M_{{\Gamma }_i}}& ={\zeta }_{M_{{\tau }_j}}\mathrm{or}{\zeta }_{M_{{\Gamma }_i}}={\zeta }_{M_{{\tau }_{j+1}}},\hfill \end{array}$$

when ${\tau }_j<{\Gamma }_i<{\tau }_{j+1}$. In a similar manner, ${\alpha }_{M_{{\Gamma }_i}}={\alpha }_{M_{{\tau }_1}}$, ${\beta }_{M_{{\Gamma }_i}}={\beta }_{M_{{\tau }_1}}$ and ${\zeta }_{M_{{\Gamma }_i}}={\zeta }_{M_{{\tau }_1}}$ when $0<{\Gamma }_i<{\tau }_1$ and ${\alpha }_{M_{{\Gamma }_i}}={\alpha }_{M_{{\tau }_p}}$, ${\beta }_{M_{{\Gamma }_i}}={\beta }_{M_{{\tau }_p}}$ and ${\zeta }_{M_{{\Gamma }_i}}={\zeta }_{M_{{\tau }_p}}$ when ${\tau }_p<{\Gamma }_i<1$.

Thus, the chain constructed in (2) is in fact contained within the chain of (1). Consequently, the two chains actually coincide, meaning there exists exactly one maximal chain. □

Remark 3.

Let $\mathfrak{F}$ be a free $\mathfrak{R}$-module and S one of its sub-modules, where $\mathfrak{R}$ is assumed to be a principal ideal domain. By [13], under this condition, the rank of a sub-module cannot exceed that of the ambient module; in particular, if $S⊊\mathfrak{F}$, then $rank\left(S\right)<rank\left(\mathfrak{F}\right)$. Since every nonempty τ-level subset of a neutrosophic sub-module forms an ordinary sub-module of $\mathfrak{F}$, each of these levels, such as ${\mathcal{M}}_{M{\tau }_1}$ and ${\mathcal{M}}_{M{\tau }_2}$, inherits a well-defined rank as a free $\mathfrak{R}$-module. Consequently, the presence of a strict inclusion between two such neutrosophic levels, i.e.,

$${\mathcal{M}}_{M{\tau }_1}⊊{\mathcal{M}}_{M{\tau }_2},$$

implies a corresponding strict inequality between their ranks:

$$rank\left({\mathcal{M}}_{M{\tau }_1}\right)<rank\left({\mathcal{M}}_{M{\tau }_2}\right).$$

This establishes that proper containment among neutrosophic sub-modules enforces a strict decrease in the rank of their associated classical modules.

Theorem 3.

Presume ${\mathcal{M}}_M$ is a neutrosophic finitely generated free $\mathfrak{R}$-module. Then, there exists a unique sequence of real numbers

$$\{{\tau }_i^{\prime }\mid i=0,1,2,\dots ,p\},$$

with $0<{\tau }_0^{\prime }<{\tau }_1^{\prime }<\dots <{\tau }_p^{\prime }<1$, satisfying the following conditions:

(a) The chain

$$\begin{array}{cc}\hfill {\alpha }_{M_1}& \le {\alpha }_{M_{{\tau }_p^{\prime }}}\le \dots \le {\alpha }_{M_{{\tau }_2^{\prime }}}\le {\alpha }_{M_{{\tau }_1^{\prime }}}\le {\alpha }_{M_0},\hfill \\ \hfill {\beta }_{M_1}& \ge {\beta }_{M_{{\tau }_p^{\prime }}}\ge \dots \ge {\beta }_{M_{{\tau }_2^{\prime }}}\ge {\beta }_{M_{{\tau }_1^{\prime }}}\ge {\beta }_{M_0},\hfill \\ \hfill {\zeta }_{M_1}& \ge {\zeta }_{M_{{\tau }_p^{\prime }}}\ge \dots \ge {\zeta }_{M_{{\tau }_2^{\prime }}}\ge {\zeta }_{M_{{\tau }_1^{\prime }}}\ge {\zeta }_{M_0},\hfill \end{array}$$

(3)

coincides with chain (1).

$\left(b\right)$ For each i, we define

$$\left\{\begin{array}{c}{\alpha }_{M_{{\tau }_p^{\prime }}}=\{\theta \in M\mid \alpha \left(\theta \right)=1,\alpha \left(\theta \right)={\tau }_p^{\prime }\},\\ {\beta }_{M_{{\tau }_p^{\prime }}}=\{\theta \in M\mid \beta \left(\theta \right)=1,\beta \left(\theta \right)={\tau }_p^{\prime }\},\\ {\zeta }_{M_{{\tau }_p^{\prime }}}=\{\theta \in M\mid \zeta \left(\theta \right)=1,\zeta \left(\theta \right)={\tau }_p^{\prime }\},\end{array}\right.$$

$$\left\{\begin{array}{c}{\alpha }_{M_{{\tau }_{p-1}^{\prime }}}=\{\theta \in M\mid \alpha \left(\theta \right)=1,\alpha \left(\theta \right)={\tau }_p^{\prime },\alpha \left(\theta \right)={\tau }_{p-1}^{\prime }\},\\ {\beta }_{M_{{\tau }_{p-1}^{\prime }}}=\{\theta \in M\mid \beta \left(\theta \right)=1,\beta \left(\theta \right)={\tau }_p^{\prime },\beta \left(\theta \right)={\tau }_{p-1}^{\prime }\},\\ {\zeta }_{M_{{\tau }_{p-1}^{\prime }}}=\{\theta \in M\mid \zeta \left(\theta \right)=1,\zeta \left(\theta \right)={\tau }_p^{\prime },\zeta \left(\theta \right)={\tau }_{p-1}^{\prime }\}.\end{array}\right.$$

More generally,

$$\left\{\begin{array}{c}{\alpha }_{M_{{\tau }_1^{\prime }}}=\{\theta \in M\mid \alpha \left(\theta \right)=1,\alpha \left(\theta \right)={\tau }_p^{\prime },\alpha \left(\theta \right)={\tau }_{p-1}^{\prime },\dots ,\alpha \left(\theta \right)={\tau }_1^{\prime }\},\\ {\beta }_{M_{{\tau }_1^{\prime }}}=\{\theta \in M\mid \beta \left(\theta \right)=1,\beta \left(\theta \right)={\tau }_p^{\prime },\beta \left(\theta \right)={\tau }_{p-1}^{\prime },\dots ,\beta \left(\theta \right)={\tau }_1^{\prime }\},\\ {\zeta }_{M_{{\tau }_1^{\prime }}}=\{\theta \in M\mid \zeta \left(\theta \right)=1,\zeta \left(\theta \right)={\tau }_p^{\prime },\zeta \left(\theta \right)={\tau }_{p-1}^{\prime },\dots ,\zeta \left(\theta \right)={\tau }_1^{\prime }\},\end{array}\right.$$

while

$$\left\{\begin{array}{c}{\alpha }_{M_{0^{\prime }}}=\{\theta \in M\mid \alpha \left(\theta \right)=1,\alpha \left(\theta \right)={\tau }_p^{\prime },\alpha \left(\theta \right)={\tau }_{p-1}^{\prime },\dots ,\alpha \left(\theta \right)={\tau }_1^{\prime },\alpha \left(\theta \right)={\tau }_0^{\prime }\},\\ {\beta }_{M_{0^{\prime }}}=\{\theta \in M\mid \beta \left(\theta \right)=1,\beta \left(\theta \right)={\tau }_p^{\prime },\beta \left(\theta \right)={\tau }_{p-1}^{\prime },\dots ,\beta \left(\theta \right)={\tau }_1^{\prime },\beta \left(\theta \right)={\tau }_0^{\prime }\},\\ {\zeta }_{M_{0^{\prime }}}=\{\theta \in M\mid \zeta \left(\theta \right)=1,\zeta \left(\theta \right)={\tau }_p^{\prime },\zeta \left(\theta \right)={\tau }_{p-1}^{\prime },\dots ,\zeta \left(\theta \right)={\tau }_1^{\prime },\zeta \left(\theta \right)={\tau }_0^{\prime }\}.\end{array}\right.$$

Proof. 

Since M is a free module, each neutrosophic sub-module ${\mathcal{M}}_{M_{\tau }}$ of the chain (1) is itself a neutrosophic free module. Hence, by Corollary 1, there exists a neutrosophic homogeneous basis $\{{\mathfrak{x}}_{li}\mid i=1,2,\dots ,n\}$, where $n=rank\left(M_{{\tau }_l}\right)$.

Define

$$\begin{array}{cc}\hfill {\tau }_l^{\prime }={\alpha }_1\left({\mathfrak{x}}_{l1}\right)& ={\alpha }_1\left({\mathfrak{x}}_{l2}\right)=\dots ={\alpha }_1\left({\mathfrak{x}}_{ln}\right)=min\{{\alpha }_1\left(\theta \right)\mid \theta \in M_{{\tau }_l}\},\hfill \\ \hfill {\tau }_l^{\prime }={\beta }_1\left({\mathfrak{x}}_{l1}\right)& ={\beta }_1\left({\mathfrak{x}}_{l2}\right)=\dots ={\beta }_1\left({\mathfrak{x}}_{ln}\right)=max\{{\beta }_1\left(\theta \right)\mid \theta \in M_{{\tau }_l}\},\hfill \\ \hfill {\tau }_l^{\prime }={\zeta }_1\left({\mathfrak{x}}_{l1}\right)& ={\zeta }_1\left({\mathfrak{x}}_{l2}\right)=\dots ={\zeta }_1\left({\mathfrak{x}}_{ln}\right)=max\{{\zeta }_1\left(\theta \right)\mid \theta \in M_{{\tau }_l}\}.\hfill \end{array}$$

Clearly, ${\tau }_l^{\prime }\ge {\tau }_l$. Assume ${\tau }_l^{\prime }\ge {\tau }_{l+1}$. Then we would obtain

$$\begin{array}{cc}\hfill {\alpha }_{M_{{\tau }_l^{\prime }}}& \le {\alpha }_{M_{{\tau }_{l+1}}}\le {\alpha }_{M_{{\tau }_l}},\hfill \\ \hfill {\beta }_{M_{{\tau }_l^{\prime }}}& \ge {\beta }_{M_{{\tau }_{l+1}}}\ge {\beta }_{M_{{\tau }_l}},\hfill \\ \hfill {\zeta }_{M_{{\tau }_l^{\prime }}}& \ge {\zeta }_{M_{{\tau }_{l+1}}}\ge {\zeta }_{M_{{\tau }_l}}.\hfill \end{array}$$

On the other hand, if $\theta ={\sum }_{l=1}^n{\mathfrak{r}}_i{\mathfrak{x}}_{li}\in M_{{\tau }_l}$ with ${\mathfrak{r}}_l\in \mathfrak{R}$, then

$$\begin{array}{cc}\hfill \alpha \left(\theta \right)=\alpha \left(\sum _{l=1}^n{\mathfrak{r}}_i{\mathfrak{x}}_{li}\right)& \ge min\{\alpha \left({\mathfrak{r}}_i{\mathfrak{x}}_{li}\right)\mid i=1,2,\dots ,n\}\hfill \\ & \ge min\{\alpha \left({\mathfrak{x}}_{li}\right)\mid i=1,2,\dots ,n\}={\tau }_l^{\prime },\hfill \\ \hfill \beta \left(\theta \right)=\beta \left(\sum _{l=1}^n{\mathfrak{r}}_i{\mathfrak{x}}_{li}\right)& \le max\{\beta \left({\mathfrak{r}}_i{\mathfrak{x}}_{li}\right)\mid i=1,2,\dots ,n\}\hfill \\ & \le max\{\beta \left({\mathfrak{x}}_{li}\right)\mid i=1,2,\dots ,n\}={\tau }_l^{\prime },\hfill \\ \hfill \zeta \left(\theta \right)=\zeta \left(\sum _{l=1}^n{\mathfrak{r}}_i{\mathfrak{x}}_{li}\right)& \le max\{\zeta \left({\mathfrak{r}}_i{\mathfrak{x}}_{li}\right)\mid i=1,2,\dots ,n\}\hfill \\ & \le max\{\zeta \left({\mathfrak{x}}_{li}\right)\mid i=1,2,\dots ,n\}={\tau }_l^{\prime }.\hfill \end{array}$$

Thus, $\theta \in M_{{\tau }_l^{\prime }}$, and consequently ${\alpha }_{M_{{\tau }_l^{\prime }}}\ge {\alpha }_{M_{{\tau }_l}}$, ${\beta }_{M_{{\tau }_l^{\prime }}}\le {\beta }_{M_{{\tau }_l}}$ and ${\zeta }_{M_{{\tau }_l^{\prime }}}\le {\zeta }_{M_{{\tau }_l}}$, which is a contradiction. Therefore, ${\tau }_l^{\prime }={\tau }_l$ and

$$\begin{array}{cc}\hfill {\alpha }_{M_{{\tau }_l}}& \ge {\alpha }_{M_{{\tau }_l^{\prime }}}\ge {\alpha }_{M_{{\tau }_{l+1}}},\hfill \\ \hfill {\beta }_{M_{{\tau }_l}}& \le {\beta }_{M_{{\tau }_l^{\prime }}}\le {\beta }_{M_{{\tau }_{l+1}}},\hfill \\ \hfill {\zeta }_{M_{{\tau }_l}}& \le {\zeta }_{M_{{\tau }_l^{\prime }}}\le {\zeta }_{M_{{\tau }_{l+1}}}.\hfill \end{array}$$

Actually, we have ${\alpha }_{M_{{\tau }_l}}={\alpha }_{M_{{\tau }_l^{\prime }}}$, ${\beta }_{M_{{\tau }_l}}={\beta }_{M_{{\tau }_l^{\prime }}}$ and ${\zeta }_{M_{{\tau }_l}}={\zeta }_{M_{{\tau }_l^{\prime }}}$. Thus, for each $l=1,2,\dots ,p$, one may select ${\tau }_l^{\prime }$ in place of ${\tau }_l$.

In the same manner, a value $0\le {\tau }_0^{\prime }<{\tau }_l$ can be taken rather than zero. More precisely,

$$\begin{array}{cc}\hfill {\tau }_0^{\prime }& =min\left\{\alpha \right(\theta \left)\right|\theta \in M\}\hfill \\ & =max\left\{\beta \right(\theta \left)\right|\theta \in M\}\hfill \\ & =max\left\{\zeta \right(\theta \left)\right|\theta \in M\}.\hfill \end{array}$$

Consequently, chain (3) is constructed, and it aligns exactly with chain (1). Summarizing, the membership distribution of ${\alpha }_M,{\beta }_M,\mathrm{and}{\zeta }_M$ is concentrated on the discrete set $\{{\tau }_l^{\prime }\mid l=0,1,\dots ,p\}\cup \left\{1\right\}$. Therefore, every member of the sequence (3) appears in the same structure as described in part (b) of the theorem. □

4. Conclusions

This paper introduces a novel framework for neutrosophic modules, an algebraic construction formed by superimposing neutrosophic set theory onto classical module theory. Within this framework, a new class of modules was formulated, and the mechanisms governing the development of their finitely generated forms were systematically analyzed. A special subclass, in which the neutrosophic values are restricted to a finite domain, was also examined, leading to several partial characterizations that enhance understanding of their internal structure. These findings extend the theoretical boundaries of neutrosophic algebra and provide a deeper connection between uncertainty modelling and algebraic reasoning.

Future investigations can proceed along multiple promising directions. One potential line of research involves examining the categorical behavior of these modules and clarifying their connections with established algebraic frameworks. Another direction concerns the practical use of neutrosophic modules in decision-making contexts where uncertainty, inconsistency, and indeterminacy are essential factors. Furthermore, extending the current study to encompass neutrosophic rough sets and neutrosophic hyperstructures may yield a richer theoretical perspective and broaden the range of applications.