Single Valued Neutrosophic Subrings and Ideals

AdvancesinMathematics:ScientificJournal 9 (2020),no.3,725–737 ISSN:1857-8365(printed);1857-8438(electronic) https://doi.org/10.37418/amsj.9.3.2

SINGLEVALUEDNEUTROSOPHICSUBRINGSANDIDEALS

H.JUDEIMMACULATE1 ,P.EVANZALINEBENANJAR,ANDK.SIVARANJANI

ABSTRACT.Singlevaluedneutrosophicsetalsoknownasthesubclassofneutrosophicsetisoneofthemightytooltodealwithinexact,indefinite,unsure information,vaguenessetc.Inthisstudywehaveprolongedthetheoryof neutrosophicringstoSVNSandformedsinglevaluedneutrosophicrings.We havealsoprolongedneutrosophicidealtheoryforneutrosophicidealovera ringtosinglevaluedneutrosophicidealofasinglevaluedneutrosophicring. Examplesaregiveninordertoillustratethetheoryandpartofsinglevalued neutrosophicnotationused.

1.INTRODUCTION

TheextensionofclassicalsettheorywhichisfuzzysetdealingwithvaguenesswasgivenbyZadeh[23]allowsthegradualassessmentofmembership functionvaluedintheinterval [0, 1] whereasclassicalsettheoryallowsthe membershipofelementsinthesetinbinaryterms.Inmanyrealworldproblemsclassicalsettheoryfailswithunreliableandincompleteinformationso itisnecessaryforthesystemtobedesignedinsuchawaytocopewithsuch data.Fuzzylogicinherentlyhandleimprecisedatabutsometimeswithmembershipfunctionitisuncertainandhardtobedefinedbycrispvalues.IntuitionisticfuzzysetwasintroducedasageneralizationoffuzzysetsbyK.

1correspondingauthor

2010 MathematicsSubjectClassification. 03B99,03E99,20K27.

Keywordsandphrases. Singlevaluedneutrosophicrings,singlevaluedneutrosophicideals, singlevaluedneutrosophicsubrings. 725

Atanassov(1983)[7]extendsboththeconceptsbyassessmentoftheelements bytwofunctionsnamelymembershipandnonmembershipfunctionvalued intheinterval [0, 1] Neutrosophicetymologicallyfromneutrosophycommonly knownasknowledgeofneutralthoughtrepresentsthedistinguishofneutrosophicfromfuzzy,intuitionisticfuzzysetwasintroducedbySamarandachein 1995.Itiscompressedwiththebasicindeterminacymembership,truthmembershipandfalsitymembershipwhichplaysavitalroleinmostsituation.In manyreallifeproblemsinordertopavesolutionthesystemaredesignedwith neutrosophiclogic.Thesubclassofneutrosophicsetalsoknownasanextensionofintuitionisticfuzzyset,singlevaluedneutrosophicsetwasgivenby Wang[19]whichgivesadditionalpossibilitytorepresentindefinite,unsure andvagueness.

Someknownalgebraicstructureintheliteratureincludeneutrosophicgroups, semigroups,rings,fields,neutrosophicmodulesetc.Rosenfield[16]formulatedtheconceptoffuzzysubgroupofagroupin1971whichwasredefinedbyAnthonyandSherwood[3]in1979.ChakroborthyandKhare[8] gavefuzzyhomomorphismbetweengroupsandstudieditseffectonfuzzy subgroup.TheconceptoffuzzysubringandidealwasintroducedbyWang JinLiu(1982)[19]whofuzzifiedcertainstandardresultsbasedonringsand ideal.MukherjeeandSen[13]discussedfuzzyprimeideals.JunKimand Yon[9,21]appliedtheconceptofintuitionisticfuzzysetstoidealsofanear ring.Intuitionisticfuzzysubringsofintutionisticfuzzyidealsofaringisdefinedin[4,6].IntuitionisticfuzzyringanditshomomorphismhasbeeninvestigatedbyYan[10].AftertheintroductionofneutrosophicsetsbySmarandache[18]severalauthors[1,2,14,22]haveappliedthenotionofneutrosophicsetstogrouptheory.PabitrakumarMaji[15]hadcombinedtheneutrosophicsetwithsoftsetsandintroducedanewmathematicalmodelneutrosophicsoftset.I.Arockiaraniet.al.definedthenotionoffuzzyneutrosophic sets[4].I.ArockiaraniandI.R.Sumathiintroducedfuzzyneutrosophicgroups [6].J.MartinaJency,I.Arockiaraniintroducedfuzzyneutrosophicsubgroupiods[11].Inthispaperweintroducetheconceptofsinglevaluedneutrosophic subringswealsoextendneutrosophicidealtheorytoformneutrosophicideal overaringandsinglevaluedneutrosophicidealofasinglevaluedneutrosophic ring.

2.PRELIMINARIES

Definition2.1. [20]LetXbeaspaceofpoints(objects),withagenericelementin Xdenotedbyx.Asinglevaluedneutrosophicset(SVNS)AinXischaracterizedby truth-membershipfunction TA,indeterminacy-membershipfunction IA andfalsity -membershipfunction FA.ForeachpointxinX TA(x),IA(x),FA(x) ∈ [0, 1].

WhenXiscontinuous,aSVNSAcanbewrittenas: A = X

T (x),I (x),F (x) /x,x ∈ X.

WhenXisdiscrete,aSVNSAcanbewrittenas: A = n i=1 T (xi),I (xi),F (xi) /xi,xi ∈ X.

Definition2.2. [5]AFuzzyneutrosophicsetAontheuniverseofdiscourseXis definedas A = x,TA(x),IA(x),FA(x) ,x ∈ X where T,I,F : X → [0, 1] and 0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3 .

Definition2.3. [5]LetXbeanon-emptyset,and A = x,TA(x),IA(x),FA(x) ,x ∈ X, B = x,TB (x),IB (x),FB (x) ,x ∈ X.

(1) A ⊆ B forallxif TA(x) ≤ TB (x) ,IA(x) ≤ IB (x),FA(x) ≥ FB (x)

(2) A ∪ B = x,max(TA(x),TB (x)),max(IA(x),IB (x)),min(FA(x),FB (x))

(3) A ∩ B = x,min(TA(x),TB (x)),min(IA(x),IB (x)),max(FA(x),FB (x))

(4) A\B(x)= x,min(TA(x),FB (x)),min(IA(x), 1 IB (x)),max(FA(x),TB (x)) .

Definition2.4. [5]AfuzzyneutrosophicsetAovertheuniverseXissaidtobe anulloremptyfuzzyneutrosophicsetif TA(x)=0,IA(x)=0,FA(x)=1 forall x ∈ X.Itisdenotedby 0N .

Definition2.5. [5]AfuzzyneutrosophicsetAovertheuniverseXissaidtobe absolute(universe)fuzzyneutrosophicsetif TA(x)=1,IA(x)=1,FA(x)=0 for all x ∈ X.Itisdenotedby 1N .

Definition2.6. [5]ThecomplementofafuzzyneutrosophicsetAisdenoted by Ac andisdefinedas Ac = x,TAc (x),IAc (x),FAc (x) where TAc (x)= FA(x), IAc (x)=1 IA(x),FAc (x)= TA(x).Thecomplementofafuzzyneutrosophicset Acanalsobedefinedas Ac =1N A.

Definition2.7. [6]Let (X,.) beagroupandletAbeafuzzyneutrosophicsetin X.ThenAiscalledafuzzyneutrosophicgroup(inshort,FNG)inXifitsatisfiesthe followingcondition:

(i) TA(xy) ≥ TA(x)∧TA(y),IA(xy) ≥ IA(x)∧IA(y),FA(xy) ≤ FA(x)∨FA(y) (ii) TA(x 1) ≥ TA(x),IA(x 1) ≥ IA(x),FA(x 1) ≤ FA(x)

Definition2.8. [17]Let (X,.) beagroupoidandletAandBbetwofuzzyneutrosophicsetsinX.ThenthefuzzyneutrosophicproductofAandB, A ◦ B,is definedasfollows,forany x ∈ X : TA◦B (x)=    ∨yz=x[TA(y) ∧ TB (z)]foreach(y,z) ∈ X × Xwith, yz = x 0 otherwise IA◦B (x)=  

∨yz=x[IA(y) ∧ IB (z)]foreach(y,z) ∈ X × Xwith, yz = x 0 otherwise FA◦B (x)=    ∧yz=x[FA(y) ∨ FB (z)]foreach(y,z) ∈ X × Xwith, yz = x 0 otherwise

Definition2.9. [11]Let (G,.) beagroupoidandlet 0N = A ∈ FNS(G).ThenA iscalledafuzzyneutrosophicsubgroupoidinG(inshort,FNSGPinG)if A ◦ A ⊂ A.

Definition2.10. [11]Let (G,.) beagroupoidandlet A ∈ FNS.ThenAiscalled afuzzyneutrosophicsubgroupoidinG(inshort,FNSGPinG)ifforany x,y ∈ G,TA(xy) ≥ TA(x) ∧ TA(y), IA(xy) ≥ IA(x) ∧ IA(y) and FA(xy) ≤ FA(x) ∨ FA(y).

Itisclearthat 0N and 1N areboth FNSGPs ofG.

Definition2.11. [12]LetGbeagroupandlet A ∈ FNSGP (G).ThenAis calledafuzzyneutrosophicsubgroup(inshort,FNSG)ofGif A(x 1) ≥ A(x), (i.e.,) TA(x 1) ≥ TA(x), IA(x 1) ≥ IA(x) and FA(x 1) ≤ FA(x) foreach x ∈ G.

3.SINGLEVALUEDNEUTROSOPHICSUBRINGSANDIDEALS

Definition3.1. Let (X,.) beagroupoidandletAandBbetwosinglevalued neutrosophicsetsinX.ThenthesinglevaluedneutrosophicproductofAandB, A ◦ B,isdefinedasfollows,forany x ∈ X :

TA◦B (x)=    ∨yz=x[TA(y) ∧ TB (z)]foreach(y,z) ∈ X × Xwith,yz = x 0 otherwise

IA◦B (x)=    ∨yz=x[IA(y) ∧ IB (z)]foreach(y,z) ∈ X × Xwith,yz = x 0 otherwise

FA◦B (x)=    ∧yz=x[FA(y) ∨ FB (z)]foreach(y,z) ∈ X × Xwith,yz = x 0 otherwise

Definition3.2. Let (G,.) beagroupoidandlet A ∈ SVNS.ThenAiscalled asinglevaluedneutrosophicsubgroupoidinG(inshort,SVNSGPinG)iffor any x,y ∈ G,TA(xy) ≥ TA(x) ∧ TA(y), IA(xy) ≥ IA(x) ∧ IA(y) and FA(xy) ≤ FA(x) ∨ FA(y).

Itisclearthat 0N and 1N areboth SVNSGPs ofG.

Definition3.3. Let (R, +,.) bearingandlet 0N = A ∈ SVNS(R).ThenAis calledasinglevaluedneutrosophicsubringinR(inshortSVNSRinR)ifitsatisfies thefollowingcondition:

1

IfAisaSVNS(G)withrespecttotheoperation + (insenseofDef.3.2)

2 AisaSVNSGPwithrespecttotheoperation (inthesenseofDef.3.1)

Definition3.4. LetRbearingand A ∈ SVNS(R) thenAiscalledasinglevalued neutrosophicsubringofRifitsatisfiesthefollowingcondition.Forall x,y ∈ R

(1) TA(x y) ≥ TA(x) ∧ TA(y) IA(x y) ≥ IA(x) ∧ IA(y) FA(x y) ≤ FA(x) ∨ FA(y)

(2) TA(xy) ≥ TA(x) ∧ TA(y) IA(xy) ≥ IA(x) ∧ IA(y) FA(xy) ≤ FA(x) ∨ FA(y).

Example1. Let R = {a,b,c,d} beasetwithtwobinaryoperationasfollows + b c d e b b c d e c c b e d d d e c b e e d b c

+ b b b b b b b b b c b b b b d b b b b e b b c c

Then(R,+,.)isaring.LetSVNS A = x,TA(x),IA(x),FA(x) inRdefinedby TA(b)=0 7,TA(c)=0 6,TA(d)=0 5, TA(e)=0 2,IA(a)=0 5, IA(b)=0 4 IA(c)= IA(d)=0 3 and FA(a)=0 2, FA(b)=0 4, FA(c)=0 6 , FA(d)=0 7 then A = x,TA(x),IA(x),FA(x) isansinglevaluedneutrosophicsubringofR.

Definition3.5. LetRbearingandlet 0N = A ∈ SVNS(R) bea SVNR inR.

ThenAiscalledasinglevaluedneutrosophicleftideal(inshortSVNLI)inRif A(xy) ≥ A(y) (i.e)

(1) TA(xy) ≥ TA(y)

IA(xy) ≥ IA(y)

FA(xy) ≤ FA(y) forany x,y ∈ R

(2) singlevaluedneutrosophicrightideal(inshortSVNRI)inRif A(xy) ≥ A(x) (i.e)

TA(xy) ≥ TA(x)

IA(xy) ≥ IA(x)

FA(xy) ≤ FA(x) forany x,y ∈ R.

(3) singlevaluedneutrosophicideal(inshortSVNI)inRifitisbothan SVNLI and SVNRI inR.

Proposition3.1. Let A = TA,IA,FA ∈ SVNI (R).Then TA(0) ≥ TA(x),

IA(0) ≥ IA(x) and FA(0) ≤ FA(x) forall x ∈ R.

Proof. Forall x ∈ R

TA(0)= TA(x x) ≥ TA(x) ∧ TA(x)= TA(x)

IA(0)= IA(x x) ≥ IA(x) ∧ IA(x)= IA(x)

FA(0)= FA(x x) ≤ FA(x) ∨ FA(x)= FA(x)

Proposition3.2. Let A = TA,IA,FA ∈ SVNI (R).Then TA(x)= TA( x), IA(x)= IA( x) and FA(x)= FA( x) forall x ∈ R.

Proof. For x ∈ R,

TA(x)= TA( ( x)) ≥ TA( x) ≥ TA(x),

IA(x)= IA( ( x)) ≥ IA( x) ≥ IA(x)

FA(x)= FA( ( x)) ≤ FA( x) ≥ FA(x)

Proposition3.3. IfaSVNS A = TA,IA,FA inRsatisfiesProposition(3.4)then TA(x y)= TA(0) ⇒ TA(x)= TA(y)

IA(x y)= IA(0) ⇒ IA(x)= IA(y)

FA(x y)= FA(0) ⇒ FA(x)= FA(y) forall x,y ∈ R

Proof. Let x,y ∈ R suchthat TA(x y)= TA(0).Then

i) TA(x)= TA(x y + y) ≥ TA(x y) ∧ TA(y)

= TA(0) ∧ TA(y)= TA(y) ii) IA(x)= IA(x y + y) ≥ IA(x y) ∧ IA(y)

= IA(0) ∧ IA(y)= IA(y) iii) FA(x)= FA(x y + y) ≤ FA(x y) ∨ FA(y)

= FA(0) ∨ FA(y)= FA(y)

Similarly

i) TA(y)= TA(x x + y)= TA(x (x y))

≥ TA(x) ∧ TA(x y)= TA(x) ii) IA(y)= IA(x x + y)= IA(x (x y))

≥ IA(x) ∧ IA(x y)= IA(x) iii) FA(y)= FA(x x + y)= FA(x (x y))

≤ TA(x) ∨ FA(x y)= FA(x)

Proposition3.4. If {Aα}α∈Γ beanyfamilyofSVNIofR.Then α∈Γ Aα or α∈Γ Aα isaSVNI(SVNLI,SVNRI).

Proof. Suppose{Aα}α∈Γ beafamilyofSVNLIofRthen

1(i.) (∩TAi )(x y)= ∧TAi (x y) ≥∧(TAi (x) ∧ TAi (y)) =(∧TAi )(x) ∧ (∧TAi (y)) =(∩TAi )(x) ∧ (∩TAi )(y))

(ii.) (∩IAi )(x y)= ∧IAi (x y)

≥∧(IAi (x) ∧ IAi (y)) =(∧IAi )(x) ∧ (∧IAi (y)) =(∩IAi )(x) ∧ (∩IAi )(y))

(iii.) (∪FAi )(x y)= ∨FAi (x y)

≤∨(FAi (x) ∨ FAi (y)) =(∨FAi )(x) ∨ (∨TAi (y)) =(∪TAi )(x) ∨ (∪TAi )(y))

2(i.) (∩TAi )(xy)= ∧TAi (xy)

≥∧(TAi (x) ∨ TAi (y)) =(∩TAi )(x) ∧ (∩TAi )(y))

(ii.) (∩IAi )(xy)= ∧IAi (xy) ≥∧(IAi (x) ∨ IAi (y)) =(∩IAi )(x) ∧ (∩IAi )(y))

(iii.) (∪FAi )(xy)= ∨FAi (xy) ≤∨(FAi (x) ∧ FAi (y)) =(∪FAi )(x) ∧ (∪FAi )(y))

Bysimilarargumentwecanshowthat α∈Γ Aα isaSVNI(SVNLI,SVNRI) Hence α∈Γ Aα isaSVNIofR.

Lemma3.1. LetRbearingandlet A,B ∈ SVNS(R)

(1) If A,B ∈ SVNLI (R)(respSVNRI(R)andSVNI(R)),then A∩B ∈ SVNLI (R)(resp SVNRI(R)andSVNI(R)).

(2) If A ∈ SVNRI (R) and B ∈ SVNLI (R) then A ◦ B ⊂ A ∩ B.

Proof. (1)Suppose A,B ∈ SVNLI (R) andlet x,y ∈ R.Then i. TA∩B (x y)= TA(x y) ∧ TB (x y) ≥ [TA(x) ∧ TA(y)] ∧ [TB (x) ∧ TB (y)] = TA∩B (x) ∧ TA∩B (y) , ii. IA∩B (x y)= IA(x y) ∧ IB (x y) ≥ [IA(x) ∧ IA(y)] ∧ [IB (x) ∧ IB (y)] = IA∩B (x) ∧ IA∩B (y) , iii. FA∩B (x y)= FA(x y) ∨ FB (x y) ≤ [FA(x) ∨ FA(y)] ∨ [FB (x) ∨ FB (y)] = FA∩B (x) ∨ FA∩B (y) . Also

i. TA∩B (xy)= TA(xy) ∧ TB (xy) ≥ [TA(y) ∧ TB (y)] = TA∩B (y) ,

ii. IA∩B (xy)= IA(xy) ∧ IB (xy) ≥ [IA(y) ∧ IB (y)] = IA∩B (y) , iii. FA∩B (xy)= IA(xy) ∨ IB (xy) ≤ [FA(y) ∨ FB (y)] = FA∩B (y) Hence A ∩ B ∈ SVNLI (R).Similarly,wecaneasilyseetheremaining. (2)Let x ∈ G andsuppose A ◦ B(x)=0N ,thenthereisnothingtoprove. Suppose A ◦ B(x) =0N .Then A ◦ B(x)=[ x=yz [TA(y) ∧ TB (z)], x=yz [IA(y) ∧ IB (z)], x=yz [FA(y) ∨ FB (z)]] .

Since A ∈ FNRI (R) and B ∈ FNLI (R) wehave: TA(y) ≤ TA(yz)= TA(x),IA(y) ≤ IA(yz)= IA(x),FA(y) ≥ FA(yz)= FA(x) , and TB (z) ≤ TB (yz)= TB (x),IB (z) ≤ IB (yz)= IB (x),FB (y) ≥ FB (yz)= FB (x) Thus TA◦B (x)= x=yz [TA(y) ∧ TB (z)] ≤ TA(x) ∧ TB (x)= TA∩B (x) IA◦B (x)= x=yz [IA(y) ∧ IB (z)] ≤ IA(x) ∧ IB (x)= IA∩B (x) FA◦B (x)= x=yz [FA(y) ∨ FB (z)] ≥ FA(x) ∧ FB (x)= FA∩B (x) .

Hence A ◦ B ⊂ A ∩ B.Thiscompletestheproof.

Proposition3.5. LetRbearingandlet 0N = A ∈ SVNS(R) .ThenAisan SVNI(resp.anSVNLI,SVNRI)inRifandonlyifforany x,y ∈ R (1) TA(x y) ≥ TA(x) ∧ TA(y) IA(x y) ≥ IA(x) ∧ IA(y) FA(x y) ≤ FA(x) ∨ FA(y).

(2) TA(xy) ≥ TA(x) ∨ TA(y)

IA(xy) ≥ IA(x) ∨ IA(y) FA(xy) ≤ FA(x) ∧ FA(y). [respectively TA(xy) ≥ TA(y) ,IA(xy) ≥ IA(y) and FA(xy) ≤ FA(y)

Proposition3.6. LetRbeaskewfield(alsodivisionring)andlet 0N = A ∈ SVNS(R) .ThenAisanSVNI(SVNLI,SVNRI)ofRifandonlyif TA(x)= TA(e) ≤ TA(0) ,IA(x)= IA(e) ≤ IA(0) and FA(x)= FA(e) ≥ FA(0) forany 0 = x ∈ R,where0istheunityofRfor + and e istheunityofRfor

Proof. LetAbeneutrosophicfuzzyidealofRandlet 0 = x ∈ R.Then

TA(x)= TA(xe) ≥ TA(e),TA(e)= TA(x 1 x) ≤ TA(x)

IA(x)= IA(xe) ≥ IA(e),IA(e)= IA(x 1 x) ≤ IA(x)

FA(x)= FA(xe) ≤ FA(e),FA(e)= FA(x 1 x) ≥ FA(x) .

Thus TA(x)= TA(e),IA(x)= IA(e),FA(x)= FA(e) , and

TA(0)= TA(e e) ≥ TA(e) ∧ TA(e)= TA(e)

IA(0)= IA(e e) ≥ IA(e) ∧ IA(e)= IA(e)

FA(0)= FA(e e) ≤ FA(e) ∨ FA(e)= FA(e)

So TA(e) ≤ TA(0),IA(e) ≤ IA(0),FA(e) ≥ FA(0).Hencethenecessarycondition hold.

Supposethenecessaryconditionhold,let x ∈ R.Thenwehavethefollowing fourcases:

Case(i)Suppose x =0,y =0 and x = y.Then:

TA(x y)= TA(e) ≥ TA(x) ∧ TA(y)

IA(x y)= IA(e) ≥ IA(x) ∧ IA(y)

FA(x y)= FA(e) ≤ FA(x) ∨ FA(y)

TA(xy)= TA(e) ≥ TA(x) ∨ TA(y)

IA(xy)= IA(e) ≥ IA(x) ∨ IA(y)

FA(xy)= FA(e) ≤ FA(x) ∧ FA(y) .

Case(ii)Suppose x =0,y =0 and x = y.Then:

TA(x y)= TA(0) ≥ TA(x) ∧ TA(y)

IA(x y)= IA(0) ≥ IA(x) ∧ IA(y)

FA(x y)= FA(0) ≤ FA(x) ∨ FA(y)

TA(xy)= TA(e) ≥ TA(x) ∨ TA(y)

IA(xy)= IA(e) ≥ IA(x) ∨ IA(y)

FA(xy)= FA(e) ≤ FA(x) ∧ FA(y)

Case(iii)Suppose x =0,y =0 .Then:

TA(x y)= TA(x)= TA(e) ≥ TA(x) ∧ TA(y)

IA(x y)= IA(x)= IA(e) ≥ IA(x) ∧ IA(y)

FA(x y)= FA(x)= FA(0) ≤ FA(x) ∨ FA(y) .

TA(xy)= TA(0) ≥ TA(x) ∨ TA(y)

IA(xy)= IA(0) ≥ IA(x) ∨ IA(y)

FA(xy)= FA(0) ≤ FA(x) ∧ FA(y)

Case(iv)Similartocase(iii). InallthecasesAisafuzzyneutrosophicsetofR.

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DEPARTMENTOF MATHEMATICS

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