Some Characterizations of Neutrosophic Submodules of an R-module

[2] Let R be a commutative ring with unity. A module M over R, denoted as M_R is an abelian group with a law of composition written ‘+’and a scalar multiplication R × M → M, written (r,v) ⇝ rv, that satisfy these axioms

1v = v

[2] A submodule N of M_R is a nonempty subset of M_R that is closed under addition and scalar multiplication.

[21, 24] A neutrosophic set P of the universal set X (NS(X)) is defined as P={(η,tP(η),iP(η),fP(η)):η∈X} P = \{ (\eta ,{t_P}(\eta ),{i_P}(\eta ),{f_P}(\eta )):\eta \in X\} where t_P,i_P, f_P : X → (⁻0,1⁺). The three components t_P, i_P and f_P represent membership value (Percentage of truth), indeterminacy (Percentage of indeterminacy) and non membership value (Percentage of falsity) respectively. These components are functions of non standard unit interval (⁻0,1⁺) [18].

[10,21] If the components of a neutrosophic set P, t_P,i_P, f_P : X → [0,1], then P is known as single valued neutrosophic set(SVNS).

In this paper, we discuss about the algebraic structure M_R-module with underlying set as SVNS. For simplicity SVNS will be called neutrosophic set.

U^X denotes the set of all neutrosophic subset of X or neutrosophic power set of X.

[17, 21, 22] Let P, Q ∈ U^X. Then P is contained in Q, denoted as P ⊆ Q if and only if P(η) ⩽ Q(η) ∀ η ∈ X, this means that tP(η)≤tQ(η), iP(η)≤iQ(η), fP(η)≥fQ(η), ∀ η∈X {t_P}(\eta ) \le {t_Q}(\eta ), {i_P}(\eta ) \le {i_Q}(\eta ), {f_P}(\eta ) \ge {f_Q}(\eta ), \forall \eta \in X

[13, 19, 21] The complement of P = {(x,t_P(x),i_P(x), f_P(x) : x ∈ X} ∈ U^X is denoted by P^C and defined as P^C = {x, f_P(x), 1 − i_P(x), t_P(x) : x ∈ X}.

(P^C)^C = P where P ∈ U^X

[8, 13, 21] Let P, Q ∈ U^X

The union C = {η,t_C(η), i_C(η), f_C(η) : η ∈ X} of P and Q [17] is denoted by C = P ∪ Q where tC(η)=tP(η)∨tQ(η)iC(η)=iP(η)∨iQ(η)fC(η)=fP(η)∧fQ(η) \matrix{{{t_C}(\eta ) = {t_P}(\eta ) \vee {t_Q}(\eta )} \cr {{i_C}(\eta ) = {i_P}(\eta ) \vee {i_Q}(\eta )} \cr {{f_C}(\eta ) = {f_P}(\eta ) \wedge {f_Q}(\eta )} \cr }

[17,22] For any P = {(η,t_P(η), i_P(η), f_P(η)) : η ∈ X} ∈ U^X, the support P^* of P can be defined as P*={η∈X,tP(η)>0,iP(η)>0,fP(η)<1} {P^*} = \{ \eta \in X,{t_P}(\eta ) > 0,{i_P}(\eta ) > 0,{f_P}(\eta ) < 1\}

[1,16] Let P = {(η, t_P(η), i_P(η), f_P(η)) : η ∈ R} be an NS(R). Then P is called a neutrosophic ideal of R if it satisfies the following conditions ∀ η,θ ∈ R

t_P(η − θ) ≥ t_P(η) ∧ t_P(θ)

We denote the set of all neutrosophic ideals of R by U(R)

[8, 9] A neutrosophic subset P ∈ U^M_R is called a neutrosophic submodule of M_R if

t_P(0) = 1, i_P(0) = 1, f_P(0) = 0

We denote neutrosophic submodules over M_R using single valued neutrosophic set by U(M).

If P ∈ U(M), then the neutrosophic components of P can be denoted as (t_P(η), i_P(η), f_P(η)) ∀ η ∈ M_R.

[8] A neutrosophic subset γP = {η,t_γP(η), i_γP(η), f_γP(η) : η ∈ M_R,γ ∈ R} of M_R where P ∈ U^M defined as follows tγP(η)=∨{tP(θ):θ∈MR,η=γθ}iγP(η)=∨{iP(θ):θ∈MR,η=γθ}fγP(η)=∧{fP(θ):θ∈MR,η=γθ} \matrix{{{t_{\gamma P}}(\eta ) = \vee \{ {t_P}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \cr {{i_{\gamma P}}(\eta ) = \vee \{ {i_P}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \cr {{f_{\gamma P}}(\eta ) = \wedge \{ {f_P}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \cr }

Let P = {η,t_P(η), i_P(η), f_P(η); η ∈ M_R} ∈ U^M_R, then t_γP(γη) ≥ t_P(η), i_γP(γη) ≥ i_P(η) and f_γP(γη) ≤ f_P(η).

We have tγP(γη)=∨{tP(θ):θ∈MR,γη=γθ}≥tP(η),∀ η∈MR {t_{\gamma P}}(\gamma \eta ) = \vee \{ {t_P}(\theta ):\theta \in {M_R},\gamma \eta = \gamma \theta \} \ge {t_P}(\eta ),\forall \eta \in {M_R} Similarly i_γP(γη) ≥ i_P(η). Also fγP(γη)=∧{fP(θ):θ∈MR,γη=γθ}≤fP(η),∀ η∈MR {f_{\gamma P}}(\gamma \eta ) = \wedge \{ {f_P}(\theta ):\theta \in {M_R},\gamma \eta = \gamma \theta \} \le {f_P}(\eta ),\forall \eta \in {M_R}

[8] Let P = {η, t_P(η), i_P(η), f_P(η); η ∈ M_R} ∈ U^M_R, then −P={η,t−P(η),i−P(η),f−P(η);η∈MR}∈UMR - P = \{ \eta ,{t_{ - P}}(\eta ),{i_{ - P}}(\eta ),{f_{ - P}}(\eta );\eta \in {M_R}\} \in {U^{{M_R}}} where t−P(η)=tP(−η), i−P(η)=iP(−η), f−P(η)=fP(−η), ∀ η∈MR {t_{ - P}}(\eta ) = {t_P}( - \eta ), {i_{ - P}}(\eta ) = {i_P}( - \eta ), {f_{ - P}}(\eta ) = {f_P}( - \eta ), \forall \eta \in {M_R}

[8] If P = {η, t_P(η), i_P(η), f_P(η); η ∈ M_R} ∈ U^M_R, then 1.P = P and (−1)P = −P

[8] Let P ∈ U^M_R, then P ∈ U(M) if and only if the following properties are satisfied ∀ η,θ ∈ M_R, γ,β ∈ R i) tP(0)=1,iP(0)=1,fP(0)=0ii) tP(γη+βθ)≥tP(η)∧tP(θ), iP(γη+βθ)≥iP(η)∧iP(θ), fP(γη+βθ)≤fP(η)∨fP(θ) \matrix{{i)\;\;{t_P}(0) = 1,{i_P}(0) = 1,{f_P}(0) = 0} \cr {ii)\;\;{t_P}(\gamma \eta + \beta \theta ) \ge {t_P}(\eta ) \wedge {t_P}(\theta ), {i_P}(\gamma \eta + \beta \theta ) \ge {i_P}(\eta ) \wedge {i_P}(\theta ), {f_P}(\gamma \eta + \beta \theta ) \le {f_P}(\eta ) \vee {f_P}(\theta )} \cr }

Let P ∈ U(M). Then P^* is a neutrosophic submodule of M_R.

Given P ∈ U(M) and P^* = {η ∈ M_R, t_P(η) > 0, i_P(η) > 0, f_P(η) < 1}. Let η, θ ∈ P^*. Then tP(η)>0,iP(η)>0,fP(η)<1tP(θ)>0,iP(θ)>0,fP(θ)<1 \matrix{{{t_P}(\eta ) > 0,{i_P}(\eta ) > 0,{f_P}(\eta ) < 1} \cr {{t_P}(\theta ) > 0,{i_P}(\theta ) > 0,{f_P}(\theta ) < 1} \cr }

To prove that γη + β θ ∈ P^* where γ,β ∈ R

⇒ to prove that t_P(γη + β θ) > 0, i_P(γη + β θ) > 0, f_P(γη + β θ) < 1

Now tP(γη+βθ)≥tP(γη)∧tP(βθ)≥tP(η)∧tP(θ)>0 \matrix{{{t_P}(\gamma \eta + \beta \theta )} \hfill & \ge \hfill & {{t_P}(\gamma \eta ) \wedge {t_P}(\beta \theta )} \hfill \cr {} \hfill & \ge \hfill & {{t_P}(\eta ) \wedge {t_P}(\theta )} \hfill \cr {} \hfill & > \hfill & 0 \hfill \cr }

In the same way, we can prove the other two inequalities. Hence the proof.

Let P_i, i ∈ J be an arbitrary non empty family of U^M_R, then

∩_i∈J P_i = {η,t_∩i∈JPi (η), i_∩i∈JPi (η), f_∩i∈JPi (η) : η ∈ M_R} where t∩i∈JPi(η)=∧i∈JtPi(η)i∩i∈JPi(η)=∧i∈JiPi(η)f∩i∈JPi(η)=∨i∈JfPi(η) \matrix{{{t_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\eta )} \cr {{i_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \wedge \limits_{i \in J} {i_{{P_i}}}(\eta )} \cr {{f_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \vee \limits_{i \in J} {f_{{P_i}}}(\eta )} \cr }

Let P_i, i ∈ J be an arbitrary non empty family of U^M_R, then γ(∪_i∈J P_i) = ∪_{i∈ J}(γP_i) for γ ∈ R

Consider γ∪_i∈J P_i = {η,t_γ∪i∈JPi(η), i_γ∪i∈JPi(η), f_γ∪i∈JPi (η) : η ∈ M_R, γ ∈ R}

Now tγ∪i∈JPi(η)=∨{t∪i∈JPi(θ):θ∈MR,η=γθ}=∨{∨i∈JtPi(θ):θ∈MR,η=γθ}=∨i∈JtγPi(η)=t∪i∈JγPi(η) \matrix{{{t_{\gamma \bigcup\nolimits_{i \in J} {P_i}}}(\eta )} \hfill & { = \vee \{ {t_{\bigcup\nolimits_{i \in J} {P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \vee \{ \mathop \vee \limits_{i \in J} {t_{{P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \mathop \vee \limits_{i \in J} {t_{\gamma {P_i}}}(\eta )} \hfill \cr {} \hfill & { = {t_{\bigcup\nolimits_{i \in J} \gamma {P_i}}}(\eta )} \hfill \cr }

Similarly i_γ∪i∈JPi (η) = i_∪i∈JγPi (η)

Now fγ∪i∈JPi(η)=∧{f∪i∈JPi(θ):θ∈MR,η=γθ}=∧{∧i∈JfPi(θ):θ∈MR,η=γθ}=∧i∈JfγPi(η)=f∪i∈JγPi(η) \matrix{{{f_{\gamma \bigcup\nolimits_{i \in J} {P_i}}}(\eta )} \hfill & { = \wedge \{ {f_{\bigcup\nolimits_{i \in J} {P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \wedge \{ \mathop \wedge \limits_{i \in J} {f_{{P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \mathop \wedge \limits_{i \in J} {f_{\gamma {P_i}}}(\eta )} \hfill \cr {} \hfill & { = {f_{\bigcup\nolimits_{i \in J} \gamma {P_i}}}(\eta )} \hfill \cr }

Hence γ(∪_i∈J P_i) = ∪_{i∈ J}(γP_i) for γ ∈ R

Let P_i, i ∈ J be an arbitrary non empty family of U(M), then ∩_i∈J P_i ∈ U(M)

We have ∩_i∈J P_i = {η,t_∩i∈JPi (η), i_∩i∈JPi (η), f_∩i∈JPi (η) : η ∈ M_R} and t∩i∈JPi(0)=∧i∈JtPi(0)=1i∩i∈JPi(0)=∧i∈JiPi(0)=1f∩i∈JPi(0)=∨i∈JfPi(0)=0 \matrix{{{t_{\bigcap\nolimits_{i \in J} {P_i}}}(0) = \mathop \wedge \limits_{i \in J} {t_{{P_i}}}(0) = 1} \cr {{i_{\bigcap\nolimits_{i \in J} {P_i}}}(0) = \mathop \wedge \limits_{i \in J} {i_{{P_i}}}(0) = 1} \cr {{f_{\bigcap\nolimits_{i \in J} {P_i}}}(0) = \mathop \vee \limits_{i \in J} {f_{{P_i}}}(0) = 0} \cr }

Now t∩i∈JPi(γη+βθ)=∧i∈JtPi(γη+βθ)≥∧i∈J(tPi(η)∧tPi(θ))=[∧i∈JtPi(η)]∧[∧i∈JtPi(θ)]=t∩i∈JPi(η)∧t∩i∈JPi(θ) \matrix{{{t_{\bigcap\nolimits_{i \in J} {P_i}}}(\gamma \eta + \beta \theta )} \hfill & { = \mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\gamma \eta + \beta \theta )} \hfill \cr {} \hfill & { \ge \mathop \wedge \limits_{i \in J} ({t_{{P_i}}}(\eta ) \wedge {t_{{P_i}}}(\theta ))} \hfill \cr {} \hfill & { = [\mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\eta )] \wedge [\mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\theta )]} \hfill \cr {} \hfill & { = {t_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) \wedge {t_{\bigcap\nolimits_{i \in J} {P_i}}}(\theta )} \hfill \cr } in the same way we can derive i∩i∈JPi(η+θ)≥i∩i∈JPi(η)∧i∩i∈JPi(θ)f∩i∈JPi(η+θ)≤f∩i∈JPi(η)∨f∩i∈JPi(θ) \matrix{{{i_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta + \theta ) \ge {i_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) \wedge {i_{\bigcap\nolimits_{i \in J} {P_i}}}(\theta )} \cr {{f_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta + \theta ) \le {f_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) \vee {f_{\bigcap\nolimits_{i \in J} {P_i}}}(\theta )} \cr } Hence ∩_i∈J P_i ∈ U(M)

[20] Let P,Q ∈ U^M_R, then the sum P+Q={η,tP+Q(η),tP+Q(η),tP+Q(η):η∈MR}∈UMR P + Q = \{ \eta ,{t_{P + Q}}(\eta ),{t_{P + Q}}(\eta ),{t_{P + Q}}(\eta ):\eta \in {M_R}\} \in {U^{{M_R}}} defined as follows tP+Q(η)=∨{tP(θ)∧tQ(ϑ)|η=θ+ϑ,θ,ϑ∈MR}iP+Q(η)=∨{iP(θ)∧iQ(ϑ)|η=θ+ϑ,θ,ϑ∈MR}fP+Q(η)=∧{fP(θ)∨fB(ϑ)|η=θ+ϑ,θ,ϑ∈MR} \matrix{{{t_{P + Q}}(\eta ) = \vee \{ {t_P}(\theta ) \wedge {t_Q}(\vartheta )|\eta = \theta + \vartheta ,\theta ,\vartheta \in {M_R}\} } \cr {{i_{P + Q}}(\eta ) = \vee \{ {i_P}(\theta ) \wedge {i_Q}(\vartheta )|\eta = \theta + \vartheta ,\theta ,\vartheta \in {M_R}\} } \cr {{f_{P + Q}}(\eta ) = \wedge \{ {f_P}(\theta ) \vee {f_B}(\vartheta )|\eta = \theta + \vartheta ,\theta ,\vartheta \in {M_R}\} } \cr }

Let P_i, i ∈ J be an arbitrary family of U(M) where P_i = {η, t_Pi(η), i_Pi(η), f_Pi(η) : η ∈ M} for each i ∈ J. Then ∑i∈JPi={η,t∑i∈JPi(η),i∑i∈JPi(η),f∑i∈JPi(η):η∈MR} \sum\limits_{i \in J} {P_i} = \{ \eta ,{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ):\eta \in {M_R}\} where t∑i∈JPi(η)=∨{∧i∈JtPi(ηi):ηi∈MR,∑i∈Jηi=η} ∀η∈MRi∑i∈JPi(x)=∨{∧i∈JiPi(Mi):ηi∈MR,∑i∈Jηi=η} ∀η∈MRf∑i∈JPi(x)=∧{∨i∈JfPi(ηi):ηi∈MR,∑i∈Jηi=η} ∀η∈MR \matrix{{{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \vee \{ \mathop \wedge \limits_{i \in J} {t_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} \forall \eta \in {M_R}} \cr {{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(x) = \vee \{ \mathop \wedge \limits_{i \in J} {i_{{P_i}}}({M_i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} \forall \eta \in {M_R}} \cr {{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(x) = \wedge \{ \mathop \vee \limits_{i \in J} {f_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} \forall \eta \in {M_R}} \cr } where, in ∑i∈Jηi \mathop {\sum }\limits_{i \in J} {\eta _i} , at most finitely η_i ≠ 0.

If P,Q ∈ U(M), then P + Q ∈ U(M)

It is enough to prove P + Q satisfies the properties listed below ∀η,θ ∈ M_R,γ,β ∈ R

t_P+Q(0) = 1, i_P+Q(0) = 1, f_P+Q(0) = 0.

From the definition 3.5, property 1 is obvious because P,Q ∈ U(M).

Consider tP+Q(η)∧tP+Q(θ)=∨{tP(η1)∧tQ(η2):η=η1+η2}∧∨{tP(θ1)∧tQ(θ2):θ=θ1+θ2}≤∨{tP(γη1)∧tQ(γη2):γη=γη1+γη2}∧∨{tP(βθ1)∧tQ(βθ2):βθ=βθ1+βθ2}=∨{[tP(γη1)∧tP(βθ1)]∧[tQ(γη2)∧tQ(βθ2)]:γη=γη1+γη2,βθ=βθ1+βθ2}≤∨{tP(γη1+βθ1)∧tQ(γη2+βθ2):γη+βθ=γη1+βθ1+γη2+βθ2}≤tP+Q(γη+βθ) where γη+βθ=γ(η1+η2)+β(θ1+θ2) ∀ η1,η2,θ1,θ2∈MR \matrix{{{t_{P + Q}}(\eta ) \wedge {t_{P + Q}}(\theta )} \hfill & { = \mathop \vee \{ {t_P}({\eta _1}) \wedge {t_Q}({\eta _2}):\eta = {\eta _1} + {\eta _2}\} \wedge \mathop \vee \{ {t_P}({\theta _1}) \wedge {t_Q}({\theta _2}):\theta = {\theta _1} + {\theta _2}\} } \hfill \cr {} \hfill & { \le \mathop \vee \{ {t_P}(\gamma {\eta _1}) \wedge {t_Q}(\gamma {\eta _2}):\gamma \eta = \gamma {\eta _1} + \gamma {\eta _2}\} \wedge \mathop \vee \{ {t_P}(\beta {\theta _1}) \wedge {t_Q}(\beta {\theta _2}):\beta \theta = \beta {\theta _1} + \beta {\theta _2}\} } \hfill \cr {} \hfill & { = \mathop \vee \{ [{t_P}(\gamma {\eta _1}) \wedge {t_P}(\beta {\theta _1})] \wedge [{t_Q}(\gamma {\eta _2}) \wedge {t_Q}(\beta {\theta _2})]:\gamma \eta = \gamma {\eta _1} + \gamma {\eta _2},\beta \theta = \beta {\theta _1} + \beta {\theta _2}\} } \hfill \cr {} \hfill & { \le \mathop \vee \{ {t_P}(\gamma {\eta _1} + \beta {\theta _1}) \wedge {t_Q}(\gamma {\eta _2} + \beta {\theta _2}):\gamma \eta + \beta \theta = \gamma {\eta _1} + \beta {\theta _1} + \gamma {\eta _2} + \beta {\theta _2}\} } \hfill \cr {} \hfill & { \le {t_{P + Q}}(\gamma \eta + \beta \theta )\; where\; \gamma \eta + \beta \theta = \gamma ({\eta _1} + {\eta _2}) + \beta ({\theta _1} + {\theta _2}) \forall {\eta _1},{\eta _2},{\theta _1},{\theta _2} \in {M_R}} \hfill \cr }

Similarly, i_P+Q(γη + β θ) ≥ i_P+Q(η) ∧ i_P+Q(θ), f_P+Q(γη + β θ) ≤ f_P+Q(η) ∧ f_P+Q(θ) ⇒ P+ ∈ U(M).

Let P_i, i ∈ J be a family of neutrosophic submodules of an M_R. Then ∑i∈JPi∈U(M) \mathop {\sum }\limits_{i \in J} {P_i} \in U(M) .

For any η ∈ X, the neutrosophic point N^{η} {\hat N_{\{ \eta \} }} is defined as N^{η}(s)={(s,tN^{η},iN^{η},fN^{η}):s∈X} {\hat N_{\{ \eta \} }}(s) = \{ (s,{t_{{{\hat N}_{\{ \eta \} }}}},{i_{{{\hat N}_{\{ \eta \} }}}},{f_{{{\hat N}_{\{ \eta \} }}}}):s \in X\} where N^{η}(s)={(1,1,0)η=s(0,0,1)η≠s {\hat N_{\{ \eta \} }}(s) = \left\{ {\matrix{{(1,1,0)} & {\eta = s} \cr {(0,0,1)} & {\eta \ne s} \cr } } \right.

Let X be a non empty set. The neutrosophic point N^0 {\hat N_{\{ 0\} }} in X is defined as N^0(x)={(x,tN^0,iN^0,fN^0):x∈X} {\hat N_{\{ 0\} }}(x) = \{ (x,{t_{{{\hat N}_{\{ 0\} }}}},{i_{{{\hat N}_{\{ 0\} }}}},{f_{{{\hat N}_{\{ 0\} }}}}):x \in X\} where N^0(x)={(1,1,0)x=0(0,0,1)x≠0 {\hat N_{\{ 0\} }}(x) = \left\{ {\matrix{{(1,1,0)} & {x = 0} \cr {(0,0,1)} & {x \ne 0} \cr } } \right.

Let P ∈ U(M). P=N^0⇔P*={0} P = {\hat N_{\{ 0\} }} \Leftrightarrow {P^*} = \{ 0\}

If P=N^0 P = {\hat N_{\{ 0\} }} , and P^* = {η ∈ M_R,t_P(η) > 0, i_P(η) > 0, f_P(η) < 1} = {0}.

Conversely, if P^* = {0} ⇒ t_P(0) > 0, i_P(0) > 0, f_P(η) < 1 and t_P(η) = 0, i_P(η) = 0 and f_P(η) = 1 ∀ η ≠ 0. Therefore P(η)={(1,1,0)η=0(0,0,1)η≠0 =N^0 P(\eta ) = \left\{ {\matrix{{(1,1,0)} & {\eta = 0} \cr {(0,0,1)} & {\eta \ne 0} \cr } } \right. = {\hat N_{\{ 0\} }}

Let p = {η,t_P(η), i_P(η), f_P(η) : η ∈ M_R} ∈ U^M. Then the U(M) of M_R generated by neutrosophic set P can be denoted and defined as 〈P〉=∩{Q|P⊆Q:Q∈U(M)} \langle P\rangle = \cap \{ Q|P \subseteq Q:Q \in U(M)\}

If Q = 〈P〉, then P is called generator of Q.

Let P_i = {(η,t_Pi(η),i_Pi(η), f_Pi(η) : i ∈ J, η ∈ M_R} be an arbitrary non empty family of NS(M_R). Then 〈∪_{i∈ J}Pi〉 = ∑_i∈J P_i

By a corollary 3.4.1, we can write ∑i∈JPi={η,t∑i∈JPi(η),i∑i∈JPi(η),f∑i∈JPi(η):η∈MR}∈U(M) \sum\limits_{i \in J} {P_i} = \{ \eta ,{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ):\eta \in {M_R}\} \in U(M) where, for all η in M_R t∑i∈JPi(η)=∨{∧i∈JtPi(ηi):ηi∈MR,∑i∈Jηi=η}i∑i∈JPi(η)=∨{∧i∈JiPi(ηi):ηi∈MR,∑i∈Jηi=η}f∑i∈JPi(η)=∧{∨i∈JfPi(ηi):ηi∈MR,∑i∈Jηi=η} \matrix{{{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \vee \{ \mathop \wedge \limits_{i \in J} {t_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \cr {{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \vee \{ \mathop \wedge \limits_{i \in J} {i_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \cr {{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \wedge \{ \mathop \vee \limits_{i \in J} {f_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \cr } where, in ∑_i∈J η_i finitely η_i‘s ≠ 0

So we can conclude for all η in M_R

tPi(η)≤t∑i∈JPi(η),∀η∈MR {t_{{P_i}}}(\eta ) \le {t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),\forall \eta \in {M_R}

Hence P_i ⊆ ∑_i∈J P_i, ∀i ∈ J.

Now to prove that ∑_i∈J P_i is the least neutrosophic submodule and ∑_i∈J P_i contains all Pi′s P_i^\prime s .

Let Q = {η,t_Q(η), i_Q(η), f_Q(η) : η ∈ M_R} ∈ U(M) and P_i ⊆ Q,∀i ∈ J, which means that tPi(η)≤tQ(η), iPi(η)≤iQ(η), fPi(η)≥fQ(η) ∀ i∈J {t_{{P_i}}}(\eta ) \le {t_Q}(\eta ), {i_{{P_i}}}(\eta ) \le {i_Q}(\eta ), {f_{{P_i}}}(\eta ) \ge {f_Q}(\eta ) \forall i \in J

Let η ∈ M_R where ∑_i∈J η_i = η and only finitely ηi′s≠0 {\eta _i^\prime}s \ne 0 , then t∑i∈JPi(η)=∨{∧i∈JtPi(ηi):ηi∈MR,∑i∈Jηi=η}≤∨{∧i∈JtQ(ηi):ηi∈M,∑i∈Jηi=η}≤∨{tQ(∑i∈Jηi):ηi∈MR,∑i∈Jηi=η}=tQ(η) \matrix{{{t_{\sum\nolimits_{i \in J} {P_i}}}(\eta )} \hfill & { = \vee \{ { \wedge _{i \in J}}{t_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \hfill \cr {} \hfill & { \le \vee \{ { \wedge _{i \in J}}{t_Q}({\eta _i}):{\eta _i} \in M,\sum\limits_{i \in J} {\eta _i} = \eta \} } \hfill \cr {} \hfill & { \le \vee \{ {t_Q}(\sum\limits_{i \in J} {\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \hfill \cr {} \hfill & { = {t_Q}(\eta )} \hfill \cr }

In the same way, i_{∑i∈J Pi} (η) i_Q(η), f_{∑i∈J Pi} (η) ≥ f_Q(η).

⇒ ∑_i∈J P_i ⊆ Q. Hence ∑_i∈J P_i ∈ U(M) is the smallest one and contains all Pi′s P_i^\prime s . Therefore ∑_i∈J P_i is the smallest U(M) which contains ∪_{i∈ J}P_i ⊆ ∑_i∈J P_i. Hence 〈 ∪_{i∈ J}P_i〉 = ∑_i∈J P_i

Let C ∈ U(R) and P ∈ NS(M_R). Define the operations C ⊚ P and C ⊛ P as NS(M_R) as follows

C ⊚ P (η) = (η,t_C⊚P(η),i_C⊚P(η), f_C⊚P(η)) ∀ η ∈ M where tC⊚P(η)=∨{tC(γ)∧tP(θ):γ∈R,θ∈M,γθ=η}iC⊚P(η)=∨{iC(γ)∧iP(θ):γ∈R,θ∈M,γθ=η}fC⊚P(η)=∧{fC(γ)∨fP(θ):γ∈R,θ∈M,γθ=η} \matrix{{{t_{C\circledcirc P}}(\eta ) = \vee \{ {t_C}(\gamma ) \wedge {t_P}(\theta ):\gamma \in R,\theta \in M,\gamma \theta = \eta \} } \cr {{i_{C\circledcirc P}}(\eta ) = \vee \{ {i_C}(\gamma ) \wedge {i_P}(\theta ):\gamma \in R,\theta \in M,\gamma \theta = \eta \} } \cr {{f_{C\circledcirc P}}(\eta ) = \wedge \{ {f_C}(\gamma ) \vee {f_P}(\theta ):\gamma \in R,\theta \in M,\gamma \theta = \eta \} } \cr }

Let P ∈ U^M_R, then

∀ γ ∈ R, N^{γ}⊚P=γP {\hat N_{\{ \gamma \} }}\circledcirc P = \gamma P

(1) The neutrosophic point N^{γ} {\hat N_{\{ \gamma \} }} , for any γ ∈ R, is defined as N^{γ}(ς)={(ς,tN^{γ},iN^{γ},fN^{γ}):ς∈R} {\hat N_{\{ \gamma \} }}(\varsigma ) = \{ (\varsigma ,{t_{{{\hat N}_{\{ \gamma \} }}}},{i_{{{\hat N}_{\{ \gamma \} }}}},{f_{{{\hat N}_{\{ \gamma \} }}}}):\varsigma \in R\} where N^{γ}(ς)={(1,1,0)γ=ς(0,0,1)γ≠ς {\hat N_{\{ \gamma \} }}(\varsigma ) = \left\{ {\matrix{{(1,1,0)} & {\gamma = \varsigma } \cr {(0,0,1)} & {\gamma \ne \varsigma } \cr } } \right.

Consider N^{γ}⊚P(η)={(η,tN^{γ}⊚P(η),iN^{γ}⊚P(η),fN^{γ}⊚P(η))} {\hat N_{\{ \gamma \} }}\circledcirc P(\eta ) = \{ (\eta ,{t_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ),{i_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ),{f_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ))\} ∀ η ∈ M_R, γ ∈ R, we have tN^{γ}⊚P(η)=∨{tN^{γ}(ς)∧tP(θ):ς∈R,θ∈MR,ςθ=η}=∨{tP(θ):θ∈M,γ=η}=tγP(η) \matrix{{{t_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta )} \hfill & { = \vee \{ {t_{{{\hat N}_{\{ \gamma \} }}}}(\varsigma ) \wedge {t_P}(\theta ):\varsigma \in R,\theta \in {M_R},\varsigma \theta = \eta \} } \hfill \cr {} \hfill & { = \vee \{ {t_P}(\theta ):\theta \in M,\gamma = \eta \} } \hfill \cr {} \hfill & { = {t_{\gamma P}}(\eta )} \hfill \cr }

Similarly we get, iN^{γ}⊚P(η)=iγP(η) {i_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ) = {i_{\gamma P}}(\eta ) , fN^{γ}⊚P(η)=fγP(η) {f_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ) = {f_{\gamma P}}(\eta )

(2) Now consider, for any γ ∈ R,η ∈ M_R, 1 ≤ i ≤ n, n ∈ N tN^{γ}⊛P(η)=∨{∧i=1n(tN^{γ}(γi)∧tP(ηi):ri∈R,ηi∈M,∑i=1nγiηi=η}=∨{∧i=1ntP(ηi):ηi∈MR,γ∑i=1nηi=η} \matrix{{{t_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta )} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_{{{\hat N}_{\{ \gamma \} }}}}({\gamma _i}) \wedge {t_P}({\eta _i}):{r_i} \in R,{\eta _i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta \} } \hfill \cr {} \hfill & { = \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \hfill \cr }

Similarly we get iN^{γ}⊛P(η)=∨{∧i=1niP(ηi):ηi∈MR,γ∑i=1nηi=η}fN^{γ}⊛P(η)=∧{∨i=1nfP(ηi):ηi∈MR,γ∑i=1nηi=η} \matrix{{{i_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ) = \vee \{ \wedge _{i = 1}^n{i_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr {{f_{{{\hat N}_{\{ \gamma \} }}P}}(\eta ) = \wedge \{ \vee _{i = 1}^n{f_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr }

If P ∈ U(R) and Q ∈ U(M), then P ⊛ Q ∈ U(M)

From the definition of P ⊛ Q, we can write, for all 1 ≤ i ≤ n, n ∈ N tP⊛Q(0)=∨{∧i=1n(tP(γi)∧tQ(ηi):γi∈R,ηi∈MR,∑i=1nγiηi=0,}=1 when γi=ηi=0 ∀ i, since tP(0)≥tP(γ) ∀ γ∈R \matrix{{{t_{P\circledast Q}}(0)} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_P}({\gamma _i}) \wedge {t_Q}({\eta _i}):{\gamma _i} \in R,{\eta _i} \in {M_R},\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = 0,\} } \hfill \cr {} \hfill & { = 1\; when\; {\gamma _i} = {\eta _i} = 0 \forall i, since\; {t_P}(0) \ge {t_P}(\gamma ) \forall \gamma \in R} \hfill \cr }

Similarly i_P⊛Q(0) = 1 and f_P⊛Q(0) = 0.

Then prove that t_P⊛Q(η + θ) ≥ t_P⊛Q(η) ∧ t_P⊛Q(θ) for η,θ ∈ M_R,1 ≤ i ≤ n,n ∈ N tP⊛Q(η+θ)=∨{∧i=1n(tP(γi)∧tQ(zi):γi∈R,zi∈M,∑i=1nγizi=η+θ}≥∨{∧i=1n(tP(ςi)∧tQ(ηi+θi):ςi∈R, ηi,θi∈M,∑i=1nςi(ηi+θi)=η+θ,∀ i}≥∨{∧i=1n(tP(ςi)∧(tQ(ηi)∧tQ(θi))):ςi∈R, ηi,θi∈M,∑i=1nςi(ηi+θi)=η+θ,∀ i}=∨{∧i=1n(tP(ςi)∧(tQ(ηi))∧(tP(ςi))∧tQ(θi)):ςi∈R, ηi, θi∈M,∑i=1nςi(ηi+θi)=η+θ}≥∨{∧i=1n(tP(ςi)∧tQ(ηi)):ςi∈R,ηi∈M,∑i=1nςiηi=η}∧∨{∧i=1n(tP(ςi)∧tQ(θi)):ςi∈R,θi∈M,∑i=1nςiθi=θ}=tP⊛Q(η)∧tP⊛Q(θ) \matrix{{{t_{P\circledast Q}}(\eta + \theta )} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_P}({\gamma _i}) \wedge {t_Q}({z_i}):{\gamma _i} \in R,{z_i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{z_i} = \eta + \theta \} } \hfill \cr {} \hfill & { \ge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\eta _i} + {\theta _i}):{\varsigma _i} \in R, {\eta _i},{\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}({\eta _i} + {\theta _i}) = \eta + \theta ,\forall i\} } \hfill \cr {} \hfill & { \ge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge ({t_Q}({\eta _i}) \wedge {t_Q}({\theta _i}))):{\varsigma _i} \in R, {\eta _i},{\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}({\eta _i} + {\theta _i}) = \eta + \theta ,\forall i\} } \hfill \cr {} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge ({t_Q}({\eta _i})) \wedge ({t_P}({\varsigma _i})) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R, {\eta _i}, {\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}({\eta _i} + {\theta _i}) = \eta + \theta \} } \hfill \cr {} \hfill & { \ge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\eta _i})):{\varsigma _i} \in R,{\eta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}{\eta _i} = \eta \} \wedge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R,{\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}{\theta _i} = \theta \} } \hfill \cr {} \hfill & { = {t_{PQ}}(\eta ) \wedge {t_{P\circledast Q}}(\theta )} \hfill \cr }

Similarly we can prove that i_P⊛Q(η + θ) ≥ i_P⊛Q(η) ∧ i_P⊛Q(θ) η, θ ∈ M and f_P⊛Q(η + θ) ≤ f_P⊛Q(η) ∨ f_P⊙Q(θ) η, θ ∈ M_R.

Now for all 1 ≤ i ≤ n, n ∈ N tP⊛Q(γη)=∨{∧i=1n(tP(γi)∧tQ(ηi)):γi∈R,ηi∈MR,∑i=1nγiηi=γη}≥∨{∧i=1n(tP(γςi)∧tQ(θi)):ςi∈R,θi∈MR,γ∑i=1nςiθi=γη}[ when γ=1]≥∨{∧i=1n(tP(ςi)∧tQ(θi)):ςi∈R,θi∈MR,∑i=1nςiθi=η,1≤i≤n,n∈N}[ since P∈U(R)⇒tP(γςi)≥tP(γ)∨tP(ςi)≥tP(ςi) ]=tP⊛Q(η) \matrix{{{t_{P\circledast Q}}(\gamma \eta )} \hfill & = \hfill & { \vee \{ \wedge _{i = 1}^n({t_P}({\gamma _i}) \wedge {t_Q}({\eta _i})):{\gamma _i} \in R,{\eta _i} \in {M_R},\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \gamma \eta \} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n({t_P}(\gamma {\varsigma _i}) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R,{\theta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\varsigma _i}{\theta _i} = \gamma \eta \} [ when\; \gamma = 1]} \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R,{\theta _i} \in {M_R},\sum\limits_{i = 1}^n {\varsigma _i}{\theta _i} = \eta ,1 \le i \le n,n \in N\} } \hfill \cr {} \hfill & {} \hfill & {[ since\; P \in U(R) \Rightarrow {t_P}(\gamma {\varsigma _i}) \ge {t_P}(\gamma ) \vee {t_P}({\varsigma _i}) \ge {t_P}({\varsigma _i}) ]} \hfill \cr {} \hfill & = \hfill & {{t_{P\circledast Q}}(\eta )} \hfill \cr }

Similarly, i_P⊛Q(rη) ≥ i_P⊛Q(η) and f_P⊛Q(rη) ≤ f_P⊛Q(η).

Hence P ⊛ Q ∈ U(M).

Let P ∈ U^M_R and corresponding to P, define Q ∈ U^M_R such that Q = {η, t_Q(η), i_Q(η), f_Q(η) : η ∈ M_R} where tQ(η)={1η=0∨{∧i=1ntP(ηi):∑i=1nγiηi=η,ηi∈MR,γi∈R}otherwiseiQ(η)={1η=0∨{∧i=1niP(ηi):∑i=1nγiηi=η,ηi∈M,γi∈R}otherwisefQ(η)={0η=0∧{∨i=1nfP(ηi):∑i=1nγiηi=η,ηi∈MR,γi∈R}otherwise \matrix{{{t_Q}(\eta ) = \left\{ {\matrix{1 \hfill & {\eta = 0} \hfill \cr { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\nolimits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} } \hfill & {otherwise} \hfill \cr } } \right.} \hfill \cr {{i_Q}(\eta ) = \left\{ {\matrix{1 \hfill & {\eta = 0} \hfill\cr { \vee \{ \wedge _{i = 1}^n{i_P}({\eta _i}):\sum\nolimits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in M,{\gamma _i} \in R\} } \hfill & {otherwise} \hfill \cr } } \right.} \hfill \cr {{f_Q}(\eta ) = \left\{ {\matrix{0 \hfill & {\eta = 0} \hfill \cr { \wedge \{ \vee _{i = 1}^n{f_P}({\eta _i}):\sum\nolimits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} } \hfill & {otherwise} \hfill \cr } } \right.} \cr } where 1 ≤ i ≤ n, n ∈ N. Then Q ∈ U(M) and 〈P〉 = Q.

From the definition of Q, t_P(η) ≤ t_Q(η), i_P(η) ≤ i_Q(η) and f_P(η) ≥ f_Q(η) ∀η ∈ M_R, then P ⊆ Q. We know t_Q(0) = 1, i_Q(0) = 1 and f_Q(0) = 0. Let γ ∈ R, η ∈ M_R.

If γη = 0 then tQ(γη)=1≥tQ(η), iQ(γη)=1≥iQ(η) and fQ(γη)=0≤fQ(η) {t_Q}(\gamma \eta ) = 1 \ge {t_Q}(\eta ), {i_Q}(\gamma \eta ) = 1 \ge {i_Q}(\eta )\; and\; {f_Q}(\gamma \eta ) = 0 \le {f_Q}(\eta )

Suppose γη ≠ 0 then η ≠ 0 and ∀ 1 ≤ i ≤ n,n ∈ N tQ(γη)=∨{∧i=1ntP(ηi):∑i=1nγiηi=γη,ηi∈MR,γi∈R, }≥∨{∧i=1ntP(ηi):∑i=1nγςiηi=γη,ηi∈MR,ςi∈R}≥∨{∧i=1ntP(ηi):γ∑i=1nςiηi=γη,ηi∈MR,ςi∈R}≥∨{∧i=1ntP(ηi):∑i=1nςiηi=η,ηi∈MR,ςi∈R}(whenγ=1)=tQ(η) \matrix{{{t_Q}(\gamma \eta )} \hfill & = \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \gamma \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R, \} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n \gamma {\varsigma _i}{\eta _i} = \gamma \eta ,{\eta _i} \in {M_R},{\varsigma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\gamma \sum\limits_{i = 1}^n {\varsigma _i}{\eta _i} = \gamma \eta ,{\eta _i} \in {M_R},{\varsigma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n {\varsigma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\varsigma _i} \in R\} } \hfill \cr {} \hfill & {} \hfill & {(when\gamma = 1)} \hfill \cr {} \hfill & = \hfill & {{t_Q}(\eta )} \hfill \cr }