Some Characterizations of Neutrosophic Submodules of an R-module

AppliedMathematicsandNonlinearSciences(aop)(aop)1–14

AppliedMathematicsandNonlinearSciences

SubmissionInfo

CommunicatedbyJagdevSingh ReceivedAugust23rd2019 AcceptedFebruary27th2020 AvailableonlineDecember31st2020

Abstract

Inthispaper,weinvestigatesomenewcharacterizationsinthealgebraicnatureofneutrosophicsubmoduledefinedovera classicalmoduleusingsinglevaluedneutrosophicset.Weadditionallycharacterizedtheneutrosophicpointanddetermined thealgebraicpropertiesofneutrosophicpointusingtheoperationsdefinedonneutrosophicsubmodule.Finleyweexamined aneutrosophicsetasageneratorofaneutrosophicsubmoduleandderivedsomerelatedconcepts.

Keywords: Neutrosophicset,Neutrosophicpoint,Neutrosophicsubmodule,Neutrosophicideal,Neutrosophicsubmodulegenerated byneutrosophicset.

1 Introduction

Afuzzysetrepresentsvagueconceptsandcontextsexpressedinnaturallanguagebymeansofgradedmembershipofelementsin [0, 1] whichisintroducedbyLotfiA.Zadehin1965[14, 25].In1986,Attanassovput forwardintuitionisticfuzzysethypothesisasadelineationofasetinwhicheverysegmentiscorresponding withaparticipationgradesandnonenrollment[3].In1995,Smarandacheoutlinedneutrosophicsetinwhich eachelementofasetisrepresentedbythreedifferingtypesofmembershipvaluesandobjectiveistonarrowthe gapbetweenthevague,ambiguousandinexactrealworldsituations[5–7, 23].Neutrosophicsettheorygivesa thoroughscientificstageinwhichwispyanduncertainhypotheticalphenomenacanbemanagedbyhierarchal membershipofcomponents.

Thealgebraicstructureinpuremathematicscloningwithuncertaintyhasbeenstudiedbysomeauthors. In1971,AzrielRosenfieldbestowedaseminalpaperonfuzzysubgroupandW.J.Liudevelopedtheideaof fuzzynormalsubgroupandfuzzysubring.Consolidatingneutrosophicsethypothesiswithalgebraicstructures isarisingpatternintheregionofmathematicalresearch.In2011,Isaac.P,P.P.John[12]recognizedsome algebraicnatureofintuitionisticfuzzysubmoduleofaclassicalmodule.Neutrosophicalgebraicalstructures

*Correspondingauthor:1984binur@gmail.com ISSN2444-8656 doi:10.2478/AMNS.2020.2.00078

OpenAccess.©2020BinuR.andPaulIsaac,publishedbySciendo. ThisworkislicensedundertheCreativeCommonsAttributionalone4.0License.

BinuR.andPaulIsaacAppliedMathematicsandNonlinearSciences(aop) 1–14

anditspropertiesprovideusasolidmathematicalfoundationtoclarifyconnectedscientificideasindesigning, informationminingandeconomics.Inthispaperwediscussaboutthegeneratorsofaneutrosophicsubmodule andsomerelatedresults.

2 Preliminaries

Definition2.1. [2]Let R beacommutativeringwithunity.Amodule M over R,denotedas MR isanabelian groupwithalawofcompositionwritten‘+’andascalarmultiplication R × M → M,written (r, v) rv,that satisfytheseaxioms 1. 1v = v 2. (rs)v = r(sv) 3. (r + s)v = rv + sv 4. r(v + v )= rv + rv ∀ r, s ∈ R and v, v ‘ ∈ M.

Definition2.2. [2]Asubmodule N of MR isanonemptysubsetof MR thatisclosedunderadditionandscalar multiplication.

Definition2.3. [21, 24]AneutrosophicsetPoftheuniversalsetX(NS(X ))isdefinedas P = {(η , tP(η ), iP(η ), fP(η )) : η ∈ X } where tP, iP, fP : X → ( 0, 1+).Thethreecomponents tP, iP andfP representmembershipvalue(Percentageof truth),indeterminacy(Percentageofindeterminacy)andnonmembershipvalue(Percentageoffalsity)respectively.Thesecomponentsarefunctionsofnonstandardunitinterval ( 0, 1+) [18].

Remark2.1. [10,21]Ifthecomponentsofaneutrosophicset P, tP, iP, fP : X → [0, 1],thenPisknownassingle valuedneutrosophicset(SVNS).

Remark2.2. Inthispaper,wediscussaboutthealgebraicstructure MR-modulewithunderlyingsetasSVNS. ForsimplicitySVNSwillbecalledneutrosophicset.

Remark2.3. U X denotesthesetofallneutrosophicsubsetof X orneutrosophicpowersetof X

2. Theintersection C = {η , tC (η ), iC (η ), fC (η ) : η ∈ X } ofPandQ[17]isdenotedby C = P ∩ Q where tC (η )= tP(η ) ∧ tQ(η ) iC (η )= iP(η ) ∧ iQ(η ) fC (η )= fP(η ) ∨ fQ(η )

Definition2.7. [17,22]Forany P = {(η , tP(η ), iP(η ), fP(η )) : η ∈ X }∈ U X ,thesupport P∗ of P canbedefined as P∗ = {η ∈ X , tP(η ) > 0, iP(η ) > 0, fP(η ) < 1} .

Definition2.8. [1, 16]Let P = {(η , tP(η ), iP(η ), fP(η )) : η ∈ R} bean NS(R).Then P iscalledaneutrosophic idealof R ifitsatisfiesthefollowingconditions ∀ η , θ ∈ R

tP(η θ ) ≥ tP(η ) ∧ tP(θ )

iP(η θ ) ≥ iP(η ) ∧ iP(θ ) 3. fP(η θ ) ≤ fP(η ) ∨ fP(θ ) 4. tP(ηθ ) ≥ tP(η ) ∨ tP(θ )

iP(ηθ ) ≥ iP(η ) ∨ iP(θ ) 6. fP(ηθ ) ≤ fP(η ) ∧ fP(θ )

Remark2.4. Wedenotethesetofallneutrosophicidealsof R by U (R)

3 Neutrosophicsubmodule

Definition3.1. [8, 9]Aneutrosophicsubset P ∈ U MR iscalledaneutrosophicsubmoduleof MR if 1. tP(0)= 1, iP(0)= 1, fP(0)= 0

2. tP(η + θ ) ≥ tP(η ) ∧ tP(θ ) iP(η + θ ) ≥ iP(η ) ∧ iP(θ ) fP(η + θ ) ≤ fP(η ) ∨ fP(θ ), forallη , θ inMR

3. tP(γη ) ≥ tP(η ) iP(γη ) ≥ iP(η ) fP(γη ) ≤ fP(η ), forall η inMR, forall γ inR

Remark3.1. Wedenoteneutrosophicsubmodulesover MR usingsinglevaluedneutrosophicsetby U (M)

Remark3.2. If P ∈ U (M),thentheneutrosophiccomponentsof P canbedenotedas (tP(η ), iP(η ), fP(η ))

BinuR.andPaulIsaacAppliedMathematicsandNonlinearSciences(aop) 1–14

Proposition3.1. Let P = {η , tP(η ), iP(η ), fP(η ); η ∈ MR}∈ U MR ,then tγ P(γη ) ≥ tP(η ), iγ P(γη ) ≥ iP(η ) and fγ P(γη ) ≤ fP(η )

Proof. Wehave

tγ P(γη )= ∨{tP(θ ) : θ ∈ MR, γη = γθ }≥ tP(η ), ∀ η ∈ MR Similarly iγ P(γη ) ≥ iP(η ) .Also fγ P(γη )= ∧{ fP(θ ) : θ ∈ MR, γη = γθ }≤ fP(η ), ∀ η ∈ MR

Definition3.3. [8]Let P = {η , tP(η ), iP(η ), fP(η ); η ∈ MR}∈ U MR ,then P = {η , t P(η ), i P(η ), f P(η ); η ∈ MR}∈ U MR where t P(η )= tP( η ), i P(η )= iP( η ), f P(η )= fP( η ), ∀ η ∈ MR Proposition3.2. [8]If P = {η , tP(η ), iP(η ), fP(η ); η ∈ MR}∈ U MR ,then1.P = P and ( 1)P = P

Theorem3.1. [8]LetP ∈ U MR ,thenP ∈ U (M) ifandonlyifthefollowingpropertiesaresatisfied ∀ η , θ ∈ MR, γ , β ∈ R

i) tP(0)= 1, iP(0)= 1, fP(0)= 0 ii) tP(γη + βθ ) ≥ tP(η ) ∧ tP(θ ), iP(γη + βθ ) ≥ iP(η ) ∧ iP(θ ), fP(γη + βθ ) ≤ fP(η ) ∨ fP(θ )

Theorem3.2. LetP ∈ U (M).ThenP∗ isaneutrosophicsubmoduleofMR

Proof. Given P ∈ U (M) and P∗ = {η ∈ MR, tP(η ) > 0, iP(η ) > 0, fP(η ) < 1}.Let η , θ ∈ P∗.Then

tP(η ) > 0, iP(η ) > 0, fP(η ) < 1

tP(θ ) > 0, iP(θ ) > 0, fP(θ ) < 1

Toprovethat γη + βθ ∈ P∗ where γ , β ∈ R ⇒ toprovethat tP(γη + βθ ) > 0, iP(γη + βθ ) > 0, fP(γη + βθ ) < 1 Now

tP(γη + βθ ) ≥ tP(γη ) ∧ tP(βθ ) ≥ tP(η ) ∧ tP(θ ) > 0

Inthesameway,wecanprovetheothertwoinequalities.Hencetheproof.

Definition3.4. Let Pi, i ∈ J beanarbitrarynonemptyfamilyof U MR ,then 1. i∈J Pi = {η , t i∈J Pi (η ), i i∈J Pi (η ), f i∈J Pi (η ) : η ∈ MR} where t i∈J Pi (η )= ∧ i∈J tPi (η ) i i∈J Pi (η )= ∧ i∈J iPi (η ) f i∈J Pi (η )= ∨ i∈J fPi (η )

2. i∈J Pi = {η , t i∈J Pi (η ), i i∈J Pi (η ), f i∈J Pi (η ) : η ∈ MR} where

t i∈J Pi (η )= ∨ i∈J tPi (η ) i i∈J Pi (η )= ∨ i∈J iPi (η ) f i∈J Pi (η )= ∧ i∈J fPi (η )

Proposition3.3. Let Pi, i ∈ J beanarbitrarynonemptyfamilyof U MR ,then γ ( i∈J Pi)= i∈J (γ Pi) for γ ∈ R

Proof. Consider γ i∈J Pi = {η , tγ i∈J Pi (η ), iγ i∈J Pi (η ), fγ i∈J Pi (η ) : η ∈ MR, γ ∈ R} Now

tγ i∈J Pi (η )= ∨{t i∈J Pi (θ ) : θ ∈ MR, η = γθ }

= ∨{∨ i∈J tPi (θ ) : θ ∈ MR, η = γθ }

= ∨ i∈J tγ Pi (η )

= t i∈J γ Pi (η )

Similarly iγ i∈J Pi (η )= i i∈J γ Pi (η ) Now fγ i∈J Pi (η )= ∧{ f i∈J Pi (θ ) : θ ∈ MR, η = γθ } = ∧{∧ i∈J fPi (θ ) : θ ∈ MR, η = γθ } = ∧ i∈J fγ Pi (η ) = f i∈J γ Pi (η )

Hence γ ( i∈J Pi)= i∈J (γ Pi) for γ ∈ R

Theorem3.3. LetPi, i ∈ JbeanarbitrarynonemptyfamilyofU (M),then i∈J Pi ∈ U (M) Proof. Wehave i∈J Pi = {η , t i∈J Pi (η ), i i∈J Pi (η ), f i∈J Pi (η ) : η ∈ MR} and t i∈J Pi (0)= ∧ i∈J tPi (0)= 1 i i∈J Pi (0)= ∧ i∈J iPi (0)= 1 f i∈J Pi (0)= ∨ i∈J fPi (0)= 0 Now t i∈J Pi (γη + βθ )= ∧ i∈J tPi (γη + βθ ) ≥∧ i∈J(tPi (η ) ∧ tPi (θ )) =[ ∧ i∈J tPi (η )] ∧ [ ∧ i∈J tPi (θ )] = t i∈J Pi (η ) ∧ t i∈J Pi (θ )

inthesamewaywecanderive i i∈J Pi (η + θ ) ≥ i i∈J Pi (η ) ∧ i i∈J Pi (θ ) f i∈J Pi (η + θ ) ≤ f i∈J Pi (η ) ∨ f i∈J Pi (θ )

Hence i∈J Pi ∈ U (M)

BinuR.andPaulIsaacAppliedMathematicsandNonlinearSciences(aop) 1–14

Definition3.5. [20]Let P, Q ∈ U MR ,thenthesum

P + Q = {η , tP+Q(η ), tP+Q(η ), tP+Q(η ) : η ∈ MR}∈ U MR definedasfollows

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}

Definition3.6. Let Pi, i ∈ J beanarbitraryfamilyof U (M) where Pi = {η , tPi (η ), iPi (η ), fPi (η ) : η ∈ M} for each i ∈ J.Then ∑ i∈J Pi = {η , t ∑ i∈J Pi (η ), i ∑ i∈J Pi (η ), f ∑ i∈J Pi (η ) : η ∈ MR} where t ∑ i∈J Pi (η )= ∨{∧ i∈J tPi (ηi) : ηi ∈ MR, ∑ i∈J ηi = η }∀η ∈ MR i ∑ i∈J Pi (x)= ∨{∧ i∈J iPi (Mi) : ηi ∈ MR, ∑ i∈J ηi = η }∀η ∈ MR f ∑ i∈J Pi (x)= ∧{∨ i∈J fPi (ηi) : ηi ∈ MR, ∑ i∈J ηi = η }∀η ∈ MR where,in ∑ i∈J ηi,atmostfinitely ηi = 0.

Theorem3.4. IfP, Q ∈ U (M),thenP + Q ∈ U (M) Proof. Itisenoughtoprove P + Q satisfiesthepropertieslistedbelow ∀η , θ ∈ MR, γ , β ∈ R

1. tP+Q(0)= 1, iP+Q(0)= 1, fP+Q(0)= 0.

2. tA+B(γη + βθ ) ≥ tP+Q(η ) ∧ tP+Q(θ ), iP+Q(γη + βθ ) ≥ iP+Q(η ) ∧ iP+Q(θ ), fA+B(γη + βθ ) ≤ fP+Q(η ) ∨ fP+Q(θ )

Fromthedefinition 3.5,property1isobviousbecause 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(γη

Remark3.3. Let X beanonemptyset.Theneutrosophicpoint ˆ N{0} in X isdefinedas ˆ N{0}(x)= {(x, t ˆ N{0} , i ˆ N{0} , f ˆ N{0} ) : x ∈ X } where ˆ N{0}(x)= (1, 1, 0) x = 0 (0, 0, 1) x = 0

Theorem3.5. LetP ∈ U (M).P = ˆ N{0} ⇔ P∗ = {0}

Proof. If P = ˆ N{0},and P∗ = {η ∈ MR, tP(η ) > 0, iP(η ) > 0, fP(η ) < 1} = {0} Conversely,if P∗ = {0}⇒ tP(0) > 0, iP(0) > 0, fP(η ) < 1and tP(η )= 0, iP(η )= 0 andfP(η )= 1 ∀ η = 0. Therefore P(η )= (1, 1, 0) η = 0 (0, 0, 1) η = 0 = ˆ N{0}

4 NeutrosophicSubmoduleGeneratedbyNeutrosophicSet

Inthissectionwestudyaboutthe U (M) of MR generatedbysinglevaluedneutrosophicsetdefinedovera classicalmodule.

Definition4.1. Let p = {η , tP(η ), iP(η ), fP(η ) : η ∈ MR}∈ U M .Thenthe U (M) of MR generatedbyneutrosophicset P canbedenotedanddefinedas P = ∩{Q|P ⊆ Q : Q ∈ U (M)}

Remark4.1. If Q = P ,then P iscalledgeneratorof Q.

Theorem4.1. LetPi = {(η , tPi (η ), iPi (η ), fPi (η ) : i ∈ J, η ∈ MR} beanarbitrarynonemptyfamilyofNS(MR). Then ∪i∈J Pi = ∑i∈J Pi Proof. Byacorollary 3.4.1,wecanwrite ∑ i∈J Pi = {η , t ∑ i∈J Pi (η ), i ∑ i∈J Pi (η ), f ∑ i∈J Pi (η ) : η ∈ MR}∈ U (M) where, forall η inMR t ∑ i∈J Pi (η )= ∨{∧ i∈J tPi (ηi) : ηi ∈ MR, ∑ i∈J ηi = η } i ∑ i∈J Pi (η )= ∨{∧ i∈J iPi (ηi) : ηi ∈ MR, ∑ i∈J ηi = η } f ∑ i∈J Pi (η )= ∧{∨ i∈J fPi (ηi) : ηi ∈ MR, ∑ i∈J ηi = η } where,in ∑i∈J ηi finitely ηi ‘ s = 0 Sowecanconclude forall η inMR 1. tPi (η ) ≤ t ∑ i∈J Pi (η ), ∀η ∈ MR 2. iPi (η ) ≤ i ∑ i∈J Pi (η ), ∀η ∈ MR

BinuR.andPaulIsaacAppliedMathematicsandNonlinearSciences(aop) 1–14 3. fPi (η )

f

i∈J Pi (η ), ∀η

MR Hence Pi ⊆ ∑i∈J Pi, ∀i ∈ J. Nowtoprovethat ∑i∈J Pi istheleastneutrosophicsubmoduleand ∑i∈J Pi containsall Pi s. Let Q = {η , tQ(η ), iQ(η ), fQ(η ) : η ∈ MR}∈ U (M) and Pi ⊆ Q, ∀i ∈ J,whichmeansthat tPi (η ) ≤ tQ(η ), iPi (η ) ≤ iQ(η ), fPi (η ) ≥ fQ(η ) ∀ i ∈ J . Let η ∈ MR where ∑i∈J ηi = η andonlyfinitely ηi s = 0,then t∑i∈J Pi (η )= ∨{∧i∈JtPi (ηi) : ηi ∈ MR, ∑ i∈J ηi = η } ≤∨{∧i∈JtQ(ηi) : ηi ∈ M, ∑ i∈J ηi = η } ≤∨{tQ(∑ i∈J ηi) : ηi ∈ MR, ∑ i∈J ηi = η } = tQ(η ) Inthesameway, i∑i∈J Pi (η ) ≤ iQ(η ), f∑i∈J Pi (η ) ≥ fQ(η ). ⇒ ∑i∈J Pi ⊆ Q .Hence ∑i∈J Pi ∈ U (M) isthesmallestoneandcontainsall Pi s.Therefore ∑i∈J Pi isthesmallest U (M) whichcontains ∪i∈J Pi ⊆ ∑i∈J Pi.Hence ∪i∈J Pi = ∑i∈J Pi

Definition4.2. Let C ∈ U (R) and P ∈ NS(MR) .Definetheoperations C P and C P as NS(MR) asfollows

1. C P (η )=(η , tC P(η ), iC P(η ), fC 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 }

2. C P (η )=(η , tC P(η ), iC P(η ), fC P(η )) ∀ η ∈ M where

tC P(η )= ∨{∧n i=1(tC (γi) ∧ tP(ηi) : γi ∈ R, ηi ∈ M, n ∑ i=1 γiηi = η , 1 ≤ i ≤ n, n ∈ N}

iC A(η )= ∨{∧n i=1(iC (γi) ∧ iA(ηi) : γi ∈ γ , ηi ∈ M, n ∑ i=1 γiηi = η , 1 ≤ i ≤ n, n ∈ N} fC A(η )= ∧{∨n i=1( fC (γi) ∨ fA(ηi) : γi ∈ γ , ηi ∈ M, n ∑ i=1 γiηi = η , 1 ≤ i ≤ n, n ∈ N}

Theorem4.2. LetP ∈ U MR ,then

Proof. (1)Theneutrosophicpoint ˆ N{γ },forany γ ∈ R,isdefinedas ˆ N{γ }(ς )= {(ς , t ˆ N{γ } , i ˆ N{γ } , f ˆ N{γ } ) : ς ∈ R} where ˆ N{γ }(ς )= (1, 1, 0) γ = ς (0, 0, 1) γ = ς Consider ˆ N{γ } P(η )= {(η , t ˆ N{γ } P(η ), i ˆ N{γ } P(η ), f ˆ N{γ } P(η ))}∀ η ∈ MR, γ ∈ R,wehave tN{γ } P(η )= ∨{tN{γ } (ς ) ∧ tP(θ ) : ς ∈ R, θ ∈ MR, ςθ = η } = ∨{tP(θ ) : θ ∈ M, γ = η } = tγ P(η )

Similarlyweget, iN{γ } P(η )= iγ P(η ), fN{γ } P(η )= fγ P(η ) (2)Nowconsider,forany γ ∈ R, η ∈ MR,1 ≤ i ≤ n, n ∈ N t ˆ N{γ } P(η )= ∨{∧n i=1(t ˆ N{γ } (γi) ∧ tP(ηi) : ri ∈ R, ηi ∈ M, n ∑ i=1 γiηi = η } = ∨{∧n i=1tP(ηi) : ηi ∈ MR, γ n ∑ i=1 ηi = η }

Similarlyweget

i ˆ N{γ } P(η )= ∨{∧n i=1iP(ηi) : ηi ∈ MR, γ n ∑ i=1 ηi = η } f ˆ N{γ } P(η )= ∧{∨n i=1 fP(ηi) : ηi ∈ MR, γ n ∑ i=1 ηi = η }

Theorem4.3. IfP ∈ U (R) andQ ∈ U (M),thenP Q ∈ U (M)

Proof. Fromthedefinitionof P Q,wecanwrite,forall1 ≤ i ≤ n, n ∈ N tP Q(0)= ∨{∧n i=1(tP(γi) ∧ tQ(ηi) : γi ∈ R, ηi ∈ MR, n ∑ i=1 γiηi = 0, } = 1 when γi = ηi = 0 ∀ i, sincetP(0) ≥ tP(γ ) ∀ γ ∈ R

Similarly iP Q(0)= 1 andfP Q(0)= 0.

BinuR.andPaulIsaacAppliedMathematicsandNonlinearSciences(aop) 1–14

Thenprovethat tP Q(η + θ ) ≥ tP Q(η ) ∧ tP Q(θ ) for η , θ ∈ MR, 1 ≤ i ≤ n, n ∈ N

tP Q(η + θ )= ∨{∧n i=1(tP(γi) ∧ tQ(zi) : γi ∈ R, zi ∈ M, n ∑ i=1 γizi = η + θ }

≥∨{∧n i=1(tP(ςi) ∧ tQ(ηi + θi) : ςi ∈ R, ηi, θi ∈ M, n ∑ i=1 ςi(ηi + θi)= η + θ , ∀ i}

≥∨{∧n i=1(tP(ςi) ∧ (tQ(ηi) ∧ tQ(θi))) : ςi ∈ R, ηi, θi ∈ M, n ∑ i=1 ςi(ηi + θi)= η + θ , ∀ i} = ∨{∧n i=1(tP(ςi) ∧ (tQ(ηi)) ∧ (tP(ςi)) ∧ tQ(θi)) : ςi ∈ R, ηi, θi ∈ M, n ∑ i=1 ςi(ηi + θi)= η + θ }

≥∨{∧n i=1(tP(ςi) ∧ tQ(ηi)) : ςi ∈ R, ηi ∈ M, n ∑ i=1 ςiηi = η }∧ ∨{∧n i=1(tP(ςi) ∧ tQ(θi)) : ςi ∈ R, θi ∈ M, n ∑ i=1 ςiθi = θ }

= tP Q(η ) ∧ tP Q(θ )

Similarlywecanprovethat iP Q(η + θ ) ≥ iP Q(η ) ∧ iP Q(θ ) η , θ ∈ M and fP Q(η + θ ) ≤ fP Q(η ) ∨ fP Q(θ ) η , θ ∈ MR Nowforall1 ≤ i ≤ n, n ∈ N

tP Q(γη )= ∨{∧n i=1(tP(γi) ∧ tQ(ηi)) : γi ∈ R, ηi ∈ MR, n ∑ i=1 γiηi = γη } ≥∨{∧n i=1(tP(γςi) ∧ tQ(θi)) : ςi ∈ R, θi ∈ MR, γ n ∑ i=1 ςiθi = γη }[ when γ = 1] ≥∨{∧n i=1(tP(ςi) ∧ tQ(θi)) : ςi ∈ R, θi ∈ MR, n ∑ i=1 ςiθi = η , 1 ≤ i ≤ n, n ∈ N} [ sinceP ∈ U (R) ⇒ tP(γςi) ≥ tP(γ ) ∨ tP(ςi) ≥ tP(ςi)] = tP Q(η )

Similarly, iP Q(rη ) ≥ iP Q(η ) andfP Q(rη ) ≤ fP Q(η ). Hence P Q ∈ U (M).

Theorem4.4. LetP ∈ U MR andcorrespondingtoP,defineQ ∈ U MR suchthatQ = {η , tQ(η ), iQ(η ), fQ(η ) : η ∈ MR} where tQ(η )= 1 η = 0 ∨{∧n i=1tP(ηi) : ∑n i=1 γiηi = η , ηi ∈ MR, γi ∈ R} otherwise iQ(η )= 1 η = 0 ∨{∧n i=1iP(ηi) : ∑n i=1 γiηi = η , ηi ∈ M, γi ∈ R} otherwise fQ(η )= 0 η = 0 ∧{∨n i=1 fP(ηi) : ∑n i=1 γiηi = η , ηi ∈ MR, γi ∈ R} otherwise where 1 ≤ i ≤ n, n ∈ N.ThenQ ∈ U (M) and P = Q.

Proof. Fromthedefinitionof Q, tP(η ) ≤ tQ(η ), iP(η ) ≤ iQ(η ) andfP(η ) ≥ fQ(η ) ∀η ∈ MR, thenP ⊆ Q Weknow tQ(0)= 1, iQ(0)= 1 andfQ(0)= 0.Let γ ∈ R, η ∈ MR If γη = 0then tQ(γη )= 1 ≥ tQ(η ), iQ(γη )= 1 ≥ iQ(η ) andfQ(γη )= 0 ≤ fQ(η )

Suppose γη = 0then η = 0 and ∀ 1 ≤ i ≤ n, n ∈ N tQ(γη )= ∨{∧n i=1tP(ηi) : n ∑ i=1 γiηi = γη , ηi ∈ MR, γi ∈ R, }

≥∨{∧n i=1tP(ηi) : n ∑ i=1 γςiηi = γη , ηi ∈ MR, ςi ∈ R}

≥∨{∧n i=1tP(ηi) : γ n ∑ i=1 ςiηi = γη , ηi ∈ MR, ςi ∈ R} ≥∨{∧n i=1tP(ηi) : n ∑ i=1 ςiηi = η , ηi ∈ MR, ςi ∈ R} (whenγ = 1) = tQ(η )

Inthesamewaywecanshowthat iQ(γη ) ≥ iQ(η ) and fQ(γη ) ≤ fQ(η ) Suppose η , θ and η + θ = 0, ∀ 1 ≤ i ≤ n, n ∈ N,then tQ(η + θ )= ∨{∧n i=1tA(zi) : n ∑ i=1 γizi = η + θ , zi ∈ MR, γi ∈ R}

≥∨{∧n i=1tP(zi) : zi = ηi + θi, n ∑ i=1 γi(ηi + θi)= η + θ , ηi, θi ∈ M, γi ∈ R}

≥∨{(∧n i=1tP(ηi)) ∧ (∧n i=1tP(θi)) : zi = ηi + θi, n ∑ i=1 γiηi + n ∑ i=1 γiθi = η + θ , ηi, θi ∈ MR, γi ∈ R}

≥∨{(∧n i=1tP(ηi) : n ∑ i=1 γiηi = η , ηi ∈ MR, γi ∈ R}∧ ∨{(∧n i=1tP(θi) : n ∑ i=1 γiθi = θ , θ

5 Conclusion

Neutrosophicsubmoduleisoneofthegeneralizationsofaclassicalalgebraicstructure,module.Thestudy ofneutrosophicsubmodulegiveextrapromptitudetotheclassicalgebraicstructuresratherthanfuzzyorintuitionisticfuzzysetsbecauseoftheinvestigationofthreedifferentlevelgradedfunctionsofeachelementin [0, 1]. Thispaperhasdevelopedamethodtoidentifygeneratorof U (M) andderivedalgebraicresultswiththehelp ofsomealgebraicoperatorsasneutrosophicsets.Thisworkareoftenextendedtothegeneratorsofarbitrary nonemptyfamilyofneutrosophicsubmodulesandstructurepreservingpropertieslikeisomorphismofneutrosophicsubmodules.Neutrosophicsubmodulesprovideusasolidmathematicalfoundationtoclarifyconnected scientificideasinimageprocessing,controltheoryandeconomicscience.

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