SoftNeutrosophicModules
MikailBalandNecatiOlgun*
DepartmentofMathematics,GaziantepUniversity,Gaziantep27310,Turkey;mikailbal46@hotmail.com * Correspondence:olgun@gantep.edu.tr
Received:22October2018;Accepted:8December2018;Published:12December2018
Abstract: Inthisstudy,forthefirsttime,westudysomebasicdefinitionsofsoftneutrosophic modulesinalgebrabeinggeneralizedanditsseveralrelatedproperties,structuralcharacteristicsare investigatedwithsuitableexamples.Inthispaper,weutilizedneutrosophicsoftsetsandneutrosophic modules.Asaresult,wedefinedsoftneutrosophicmodules.Afterweaksoftneutrosophicmodules andstrongsoftneutrosophicmodulesaredescribedandillustratedbyexamples.Finallysoft neutrosophicmodulehomomorphismisdefinedandsoftneutrosophicmoduleisomorphism isexplained.
Keywords: strongsoftneutrosophicmodule;weaksoftneutrosophicmodule;softneutrosophic submodule;softneutrosophicring
1.Introduction
In1980,FlorentinSmarandanchecharacterizedthenotionofneutrosophyasanewbranchof philosophy.“Itisabranchofphilosophywhichstudiesthenature,scopeandoriginofneutralities,as wellastheirinteractionswithdifferentideationalspectra”.Itisthebaseofneutrosophiclogicthat isanextensionfuzzylogicinthatindeterminacyisinclusive.Heintroducedtheconceptof(T-truth, I-indeterminacy,andF-falsity)memberships.Neutrosophicsetisapowergeneralformalstructure whichgeneralizestheconceptoftheintuitionisticfuzzyset,intervalvaluedfuzzyset,fuzzyset,classic setetc.
Strugglewithuncertaintiesisabigprobleminmanyareaslikemedicalscience,environment, economics,socialsciences,etc.Thesekindsofproblemscannotbeovercomebyclassicalmethods, becausethereareinherentdifficultiesintheclassicalmethods.Thereasonforthesedifficultiesmay beduetotheinadequacyofthetheoriesofparameterizationtools.ForDealingwiththesekindsof difficulties,Molodtsovsuggestedanewmathematicaltoolfordealingwithuncertainties.Thisiscalled softsettheory[1].
Thealgebraicstructureofsettheoriesdealingwithuncertaintieshasalsobeenstudiedbysome authors.Smarandancheandothers,studiedneutrosophicalgebraicstructures[2–4]byusingthe ideaofneutrosophictheoryandthiswasthefirstneutrosoficationofalgebraicstructures.Some authors[2–21]havestudiedneutrosophicalgebraicstructuresandnowadays,researchersarestill interestedinthissubject.Theimportanceofthisneutrosophicstructurescontinuestoincrease.Recently, researchershavedoneanumberofsoftneutrosophicstructuresformedbycombiningsoftstructures andneutrosophicstructures[18,20,21].
Therefore,wedecidedtoinvestigatetheeffectsofthissubjectonmodulesofalgebraicstructures. Modulesarebasicstructuresinalgebraicstructuresandaregeneralizationofthevectorspaces.This topichasnotbeenstudiedyet.
Inthispaper,forthefirsttimeweintroduce‘SoftNeutrosophicModule’,‘WeakSoft-N Module’, andStrongSoft-N Module’.Finally,wedescribedthesoft-N modulehomomorphismandthesoft-N moduleisomorphisminmoduletheory,usingneutrosophicsoftsets.Thisisnewtopicinthemodule Mathematics 2018, 6,323;doi:10.3390/math6120323 www.mdpi.com/journal/mathematics
theoryandneutrosophiclogic.Therestofthepaperisorganizedasfollows.Aftertheliterature reviewinSection 1 andPreliminariesinSection 2,weintroducesoftneutrosophicmodule,weaksoft-N moduleandstrongsoft-N moduleandalsoexaminetherelationsbetweenweaksoft-N moduleand strongsoft-N moduleinSection 3.Finally,conclusionsaregiveninSection 4.
2.Preliminaries
Inthischapter,weremembersomebasicdefinitionforsoftsetsandneutrosophicmodules.These arerelatedtofuzzysets,softsets,neutrosophicsetswhicharefoundintheliterature[1,2,5,22–26].
2.1.SoftSetsTheory
Let U beaninitialuniversesetand E beasetofparameters.Thepowersetof U isdenotedby P(U)and A ⊂ E.Molodtsovdefinedthenotationofasetinthefollowingway:
Definition1. Apair(F,A)iscalledasoftsetoverU,whereFisamappinggivenby: F : A → P(U) [1]
Definition2. Fortwosoftsets(F,A)and(G,B)overageneraluniverseU,wesaythat(F,A)isasoftsubsetof (G,B)if: (i) A ⊂ B,and (ii) F(x)andG(x)areidenticalapproximations ∀ x ∈ A.Wewrite (F, A) ⊂ (G, B) (F,A)issaidtobeasoftsupersetof(G,B)if(G,B)isasoftsubsetof(F,A).Wedenoteitby(F,A) ⊃ (G,B) (see[1]).
Definition3. Twosoftsets(F,A)and(G,B)onacommonuniverseUaresaidtobesoftequalif(F,A)isasoft subsetof(G,B)and(G,B)isasoftsubsetof(F,A)(see[1]).
Definition4. Unionoftwosoftsetsof(F,A)and(G,B)overthecommonuniverseUisthesoftset(H,C), whereC = A ∪ B,and ∀ x ∈ C, H(e) = F(x), ifx ∈ A B, = (G), ifx ∈ B A, = F(x) ∪ G(x), ifx ∈ A ∩ B.
Wewrite:
(F, A) ∪ (G, B)=(H, C)(see [1])
Definition5. Intersectionoftwosoftsets(F,A)and(G,B)overacommonuniverseUisthesoftset(H,C), whereC = A ∩ B,and ∀ x ∈ C,H(x)=F(x)orG(x),(asbotharesameset).Wewrite:
(F, A) ∩ (G, B)=(H, C)(see [1]).
2.2.NeutrosophicTheory
Definition6. Let(G,*)beanygroupand < G ∪ I >= {x + yI : x, y ∈ G} beaset.Then (< G ∪ I > , ∗)= N(G) iscalledaneutrosophicgroupgeneratedbyIandGunderthebinaryoperation ∗.Iiscalledthe neutrosophicelementwiththepropertyI2 =I.Foranintegern,n+1,andnIareneutrosophicelementsand0.I =0.I 1,theinverseofIisnotdefinedandhencedoesnotexist(see[2]).
Theorem1. Let (< G ∪ I >, ∗)= N(G) betheneutrosophicgroup.Then:
(i) N(G)alwayscontainsagroup. (ii) N(G)isnotagroupingeneral(see[2]).
Definition7. Let < G ∪ I >= N(G) beaneutrosophicgroupgeneratedbyIandG.ApropersubsetP(G) issaidtobeaneutrosophicsubgroupifP(G)isaneutrosophicgroup(i.e.,P(G)shouldcontainasubgroup) (see[5]).
Definition8. LetN(G)beanyneutrosophicgroup.Ifab=baforeacha,b ∈ N(G),then(N(G),*)iscalleda commutativeneutrosophicgroup(see[5]).
Theorem2. LetN(P)andN(M)beanytwoneutrosophicsubgroupsofacommutativeneutrosophicgroup N(G).Then:
(i) N(P) ∩ N(M) isaneutrosophicsubgroupofN(G). (ii) N(P) N(M) isaneutrosophicsubgroupofN(G), (iii) N(P) ∪ N(M) isaneutrosophicsubgroupofN(G)ifandonlyif N(M) ⊆ N(P) or N(P) ⊆ N(M) (see[5]).
Definition9. Let(R,+,*)beanyring.Theset < R ∪ I >= {x + yI : x, y ∈ R} isneutrosophicring generatedbyIandRundertheoperationofR(see[5]).
Theorem3. If < R ∪ I > aneutrosophicring,then < R ∪ I > isaring(see[5]).
Definition10. Let < R ∪ I > beaneutrosophicring,Wesaythat < R ∪ I > iscommutativeneutrosophic ringifab=ba ∀a,b<R ∪ I>(see[5]).
3.SoftNeutrosophic(Soft-N)Modules
Definition11. Let((H,A),+,.)beanysoftG-module(G-mod)overacommutativesoftring(G,A)andlet(H, A)(I)=<(H,A) ∪ I>beasoft-Nsetgeneratedby(H,A)andI.Thetriple((H,A)(I),+,.)isdefinedaweak soft-NG-modoverasoftring(G,A).Incaseof(G,A)(I)isasoft-NG-modoverasoft-Nring(G,A)(I),itis calledastrongsoft-NG-mod.Theelementsof(H,A)(I)arecalledsoft-Nelementsandtheelementsof(G,A)(I) aredescribedsoft-Nscalars.
Ifa1 =k+mI,a2 =n+pI ∈ H(I)wherek,m,n,p ∈ (H,A)andq=u+vI ∈ (G,A)(I)whereu,vare scalarsinG,wedefine:
a1 +a2 =(k+mI)+(n+pI)=(k+n)+(m+p)Iand qa1 =(u+vI).(k+mI)=uk+(um+kv+vm)I.
Example1. Let(G,A)beacommutativesoftringandKbeanidealof(G,A).
(i) (G,A)(I)isasoft-NG-moditself,because(G,A)(I)isasoft-Nabeliangroup.Hencethemultiplicationin Ggivesusamapping: G × (G, A)(I) → (G, A)(I).
Thesoft-NringaxiomsprovidethatthisscalarmultiplicationturnsGintoasoft-NG-mod.
(ii) (G,A)(I)isaweaksoft-NG-modonasoftring(Q,A).Moreoverthisisastrongsoft-NG-modona soft-Nring(Q,A)(I).
(iii) IfKisclosedunderadditionandmultiplicationbyarbitraryelementsofsoft(G,A)(I),thenKisansoft-N G-modunderthe(+)and(.)ofG,
(iv) (G,A)n (I)isaweaksoft-NG-modonasoftring(G,A).Moreoverthisisastrongsoft-NG-modovera soft-Nring(G,A)(I).
(v) (H,A)m×n (I)={[aij]:aij ∈ (Q,A)(I)}isaweaksoft-NG-modonasoftring(Q,A).Moreoverthisisa strongsoft-NG-modonasoft-Nring(Q,A)(I).
Theorem4. Eachstrongsoft-NG-modisaweaksoft-NG-mod.
Proof. Let(H, A)(I)beastrongsoft-N moduleoverasoft-N ring(G, A)(I).Wehave,(G, A) ⊆ (G, A)(I) istrueforeverysoftring(G, A).Therefore(H, A)(I)isaweaksoft-NG-mod
Theorem5. Eachweak(strong)soft-NG-modisasoftG-mod.
Proof. Assumethat(H, A)(I)isastrongsoft-N moduleoverasoft-N ring(G, A)(I).Clearlyvisible, ((H, A)(I),+,.)isanabeliansoftgroup.Let a1 = k +mI, a2 = n+pI ∈ (H, A)(I), q = x+yI, w = z+tI ∈ (G, A)(I)where k,m,n,p ∈ (H, A)and x,y,z,t ∈ G.Then:
(1) q (a1 + a2)=(x + yI)(k + mI + n + pI) = xk + xn +[xm + xp + yk + ym + yn + yp]I =(x + yI)(k + mI)+(x + yI)(n + pI) = qa1+ qa2
(2)(q + w)a1 =(x + yI + z + tI)(k + mI) = xk + zk +[xm + zm + yk + tk + ym + tm]I =(x+yI)(k+mI)+(z+tI)(k+mI) = qa1 + wa1
(3)(qw) a1=((x + yI)(z+tI))(k+mI) = xzk +[xzm+xtk+yzk+ytk+xtm+yzm+ytm]I =(x+yI)((z+tI)(k+mI)) = q(wa1)
(4)For1+1+0I ∈ G(I),wehave 1a1=(1+1I)(k + mI) =1k +1mI = k + mI = a1.
Consequently,(H, A)(I)isasoft G-mod
Lemma1. Assumethat(H,A)(I)isastrongsoft-NG-modoverasoft-Nring(G,A)(I)andleta1 =k+mI,a2 =n+pI,a3 =u+vI ∈ (H,A)(I),q=x+yI ∈ (G,A)(I).Then:
(i) a1 +a3=a2 +a3 impliesa1=a2 (ii) q0=0. (ii) 0a1 =0.
(iv) ( q)a1 =q( a1)= (qa1).
Proof. (i)Let a1 + a3= a2 + a3.Then(k + mI)+(u+vI)=(n+pI)+(u+vI).Clearlyweobtain(k + mI)= (n+pI)fromdefinitionofthestrongsoft-NG-mod
(i) q(0+0I)=(x+yI)(0+0I)=(x0+x0+y0+ y0I)=(0+0I)=0 (ii) 0a1 =0(k + mI)=0
(iii) ( q) a1 =( x yI)(n+pI)=( xn xpI ynI ypI) q( a1)=(x + yI)( n pI)=( xn xpI ynI ypI) (qa1)= ((x + yI)(n + pI))=( xn xpI ynI ypI).
Henceweobtaintheequality ( q)a1 =q( a1)= (qa1).
Definition12. Let(H,A)(I)beastrongsoft-NG-modonasoft-Nring(G,A)(I)and ∅ = (N,A)(I) ⊂ (H, A)(I).If(N,A)(I)isitselfastrongsoft-NG-modover(G,A)(I),thenitiscalledastrongsoft-Nsubmoduleof (M,A)(I).Itisnecessarythat(M,A)(I)mustcontainsapropersoftsubsetwhichisasoftG-mod.
Definition13. Let(H,A)(I)beaweaksoft-NG-modoverasoftring(G,A)and ∅ = (N,A)(I) ⊂ (H,A)(I). if(N,A)(I)isitselfaweaksoft-NG-modover(G,A),thenitiscalledaweaksoft-Nsubmoduleof(H,A)(I),It’s necessarythat(N,A)(I)mustcontainsapropersubsetwhichisasoftG-mod.
Theorem6. Let(H,A)(I)beastrongsoft-NG-modonasoft-Nring(G,A)(I)and ∅ = (N,A)(I) ⊂ (H,A) (I).Then(N,A)(I)isastrongsoft-Nsubmoduleof(H,A)(I)ifandonlyifthefollowingconditionshold:
(i) a1,a2∈ (N,A)(I),impliesa1 +a2∈ (N,A)(I).
(ii) a1 ∈ (N,A)(I)andq=x+yI∈(G,A)(I)forallx,y∈Gimpliesqa1 ∈ (N,A)(I). (iii) (N,A)(I)mustincludeapropersoftsubsetwhichisasoftG-mod.
Proof. (⇒)Thisisclear,since(N,A)(I)mustcontainthezeroelement0of(H,A)(I),and(N,A)(I) mustbeclosedundertheadditionof(H,A)(I)andunderscalarmultiplicationbyarbitraryelements of(G, A)(I).
(⇐)BytheSubgroupCriterion, (N,A)(I) isanadditivesubgroupof(H, A)(I);also,by(ii), (N,A) (I) isclosedunderscalarmultiplicationbyarbitraryelementsof(G, A)(I).Itisnowclearthatmodule propertiesareautomaticallyinheritedfrom(H, A)(I).
If(i)and(ii)arecombinedintheabovetheorem,wecanwriteasfollows.
Corollary1. Let(H,A)(I)beastrongsoft-NG-modonasoft-Nring(G,A)(I)andlet ∅ = (N,A)(I) ⊂ (H, A)(I).Then(N,A)(I)isastrongsoft-Nsubmoduleof(M,A)(I)ifandonlyifthefollowingconditionshold:
(i) a1,a2 ∈ (N,A)(I)andeveryq,w ∈ (G,A)(I),impliesqa1 +wa2 ∈ (N,A)(I) (ii) (N,A)(I)mustincludeapropersoftsubsetwhichisasoftG-mod.
Example2. Let(H,A)(I)beaweak(strong)soft-NG-mod.Then(H,A)(I)isaweak(strong)soft-Nsubmodule ofitself.Itiscalledatrivialweak(strong)soft-Nsubmodule.
Example3. Let(H,A)(I)=(G,A)(I)beastrongsoft-NG-modonasoft-Nring(G,A)(I).Theset(N,A)(I)= {(a1 =x+yI,a2 =z+tI,0=0+0I) ∈ (H,A)(I):x,y,z,t ∈ (H,A)}isastrongsoft-Nsubmoduleof(H,A)(I).
Example4. Let(H,A)(I)=(H,A)m*n(I)={[xij]:xij ∈ (H,A)(I)}beastrongsoft-NG-modon(G,A)(I). Thesetofsoftmatrices(N,A)(I)=(H,A)m*n (I)={[yij]:yij ∈ (G,A)(I)andtrace(H)=0}isastrongsoft-N submoduleof(G,A)(I).
Theorem7. Assumethat(H,A)(I)isastrongsoft-NG-modonasoft-Nring(G,A)(I)and{(N,A)n(I)}n∈λ is afamilyofstrongsoft-Nsubmodulesof(H,A)(I).Then ∩ (N,A)n(I)isastrongsoft-Nsubmoduleof(H,A)(I).
Proof. Obviously,0(H,A) ∩ (N,A)n(I)for ∀n 0(M,A),A)n(I andhence ∩ (N,A)n(I) = Ø.Let x,y ∩ (N, A)n(I)andlet a (G, A)(I)soft N-ring.Dueto,for ∀nx yN,A)n(I and axN,A)n(I thenx y ∩ (N, A)n(I)and ax ∩ (N,A)n(I).Therefore ∩ (N,A)n(I)isastrongsoft-N submoduleof(H,A)(I).
Remark1. Let(H,A)(I)beastrongsoft-NG-modonasoft-Nring(G,A)(I)andlet(N,A)1(I)and(N,A)2(I) betwodistinctstrongsoft-Nsubmoduleof(H,A)(I).Usually,(N,A)1(I) ∪ (N,A)2(I)isnotastrongsoft-N submoduleof(H,A)(I).Ontheotherhand,inthatcase(N,A)1(I) ⊆ (N,A)2(I)or(N,A)2(I) ⊇ (N,A)1(I),(N, A)1(I)U(N,A)2(I)isastrongsoft-Nsubmoduleof(H,A)(I).
Definition14. Let(H,A)(I)and(N,A)(I)bestrongsoft-NG-modonasoft-Nring(G,A)(I)andlet σ:(H, A)(I) → (N,A)(I)beamappingof(H,A)(I)into(N,A)(I). σ iscalledasoft-NG-modhomomorphismbut onlyifthefollowingconditionshold:
(i) σ isasoftG-modhomomorphism. (ii) σ (I)=I.
If σ isabijectivesoft-NG-modhomomorphism, σ iscalledasoft-NG-modisomorphismanditisshownas (H,A)(I) ∼ = (N,A)(I).
Definition15. Let(H,A)(I)and(N,A)(I)bestrongsoft-NG-modonasoft-Nring(G,A)(I)andlet σ:(H, A)(I) → (N,A)(I)beasoft-NG-modhomomorphism.
(i) Theset{n(N,A)(I): σ(h)=nforsomehH,A)(I)}iscalledtheimageof σ anditisdenotedbyImσ. (ii) Theset{am (H,A)(I): σ (am)=0forsomeam H(I)}iscalledthekernelof σ anditisdenotedbyKerσ
Example5. Let(H,A)(I)beastrongsoft-NG-modoverasoft-Nring(G,A)(I).
(i) Themapping σ:(H,A)(I) → (H,A)(I)isdescribedas σ(a)=a ∀ a ∈ (H,A)(I)issoft-NG-mod homomorphism.FurthermoreKerσ =0.
(ii) Themapping σ:(H,A)(I) → (H,A)(I)isdescribedas σ(a)=0 ∀ a ∈ (H,A)(I)issoft-NG-mod homomorphismsinceI ∈ (H,A)(I).Ontheotherhand σ(I) = 0.
Theorem8. Let(H,A)(I)and(N,A)(I)bestrongsoft-NG-modonasoft-Nring(G,A)(I).If σ:(H,A)(I) → (N,A)(I)beasoft-NG-modhomomorphism,then
(i) Thekernelof σ isnotastrongsoft-Nsubmoduleof(H,A)(I),ontheotherhanditisasoftsubmoduleof (H,A).
(ii) Theimageof σ isastrongsoft-Nsubmoduleof(N,A)(I).
Proof. (i)Clearly, I (H,A)(I)and σ (I)0.Thisimpliesthatkernelof σ doesnotcontain I.Therefore Kerσ isnotastrong soft-Nsubmoduleof (H,A)(I).Butthekernelof σ isasoftsubmoduleof(H,A)[26]. (ii)Open.
Theorem9. Let(N,A)(I)beastrongsoft-Nsubmoduleofastrongsoft-NG-mod(M,A)(I)onasoft-Nring (G,A)(I).If σ:(H,A)(I) → (H,A)(I)/(N,A)(I)isamappingdescribedas σ(a)=a+(N,A)(I) ∀ a ∈ (H,A) (I)then σ isnotasoft-NG-modhomomorphism.
Proof. Wehave σ(I)=I + (N,A)(I) = (N,A)(I) = I fromdefinitionof σ.Therefore σ isnota soft-NG-mod homomorphism.
Theorem10. Let(N,A)(I)beastrongsoft-Nsubmoduleofastrongsoft-NG-mod(H,A)(I)onasoft-Nring (G,A)(I)andlet σ:(H,A)(I) → (K,A)(I)beasoft-NG-modhomomorphismon(G,A)(I).If σ(N,A)(I):(N,A) (I) → (K,A)(I)istherestrictionof σ to(N,A)(I)isdescribedas σ(N,A)(I) (n)= σ(n).Then:
(i) Ker σ ∩ (N,A)(I)=(N,A)(I)Kerσ(N,A)(I) (ii) σ (N,A)(I) isasoft-NG-modhomomorphism. (iii) σ ((N,A)(I))=Imσ(N,A)(I)
Proof. (i)Itisclearfromthehomologicalpropertiesof σ
(i) Therestrictionof σ to (N,A)(I), σ (N,A)(I) isasoft-NG-mod homomorphismsince σ isasoft-N G-mod homomorphism. (ii) Itisobtaineasilyfromdefinitionofthe σ.
Remark2. If(H,A)(I)and(N,A)(I)arestrongsoft-NG-modonasoft-Nring(G,A)(I).Let σ, µ:(H,A)(I) → (N,A)(I)aresoft-NG-modhomomorphisms.Then(σ + µ)and(kσ)arenotsoft-NG-modhomomorphisms foranyk ∈ (G,A)(I).Because(σ + µ)(I)= σ(I)+ µ(I)=I+I=2I = Iand(kσ)(I)=kσ(I)=kI = Iforeveryk ∈ (G,A)(I).
Therefore,IfHom((H,A)(I),(N,A)(I))isthecollectionofeverysoft-NG-modhomomorphismsfrom (H,A)(I)into(N,A)(I)anditisnotasoft-NG-modon(G,A)(I).Thatiscompletelydifferentfromwhichis attainableintheclassicalG-mod.
Definition16. Let(N,A)(I),(H,A)(I)and(K,A)(I)bestrongsoft-NG-modonasoft-Nring(G,A)(I).Let σ:(K,A)(I) → (H,A)(I)and µ:(H,A)(I) → (N,A)(I)besoft-NG-modhomomorphism.,thecomposition µσ: (K,A)(I) → (N,A)(I)isdescribedas µ (σ(k))= µσ(k)for ∀ k ∈ (K,A)(I).
Theorem11. Let(N,A)(I),(H,A)(I)and(K,A)(I)bestrongsoft-NG-modonasoft-Nring(G,A)(I).If σ: (K,A)(I) → (H,A)(I)and µ:(H,A)(I) → (N,A)(I)aresoft-NG-modhomomorphisms,thenthecomposition σ and µ, µσ:(K,A)(I) → (N,A)(I)isasoft-NG-modhomomorphism.
Proof. Obviously, µσ isasoft G-mod homomorphismfor kIK,A)(I)andwehave:
µσ(I)=µ(σ(I)) = µ(I) = I.
Therefore, µσ isasoft-NG-mod homomorphism.
Corollary1. LetHOM((H,A)(I),(H,A)(I))bethecollectionofeverysoft-NG-modhomomorphismsfrom(H, A)(I)onto(H,A)(I).Thenwehave (σµ)λ = σ(µλ) for ∀ λ, µ, σ, ∈ HOM((H,A)(I),(H,A)(I)).
Proof. ItisopenfromtheTheorem11.
Theorem12. Assumethat(K,A)(I),(H,A)(I)and(N,A)(I)arestrongsoft-NG-modonasoft-Nring(G,A) (I).and σ:(K,A)(I) → (H,A)(I)and µ:(H,A)(I) → (N,A)(I)aresoft-NG-modhomomorphisms.Then,
(i) If µσ issurjective,then σ issurjective.
(ii) If µσ isinjective,then µ isinjective.
(iii) If µ and σ areinjective,then µσ isinjective.
Proof. Itisclearfromthepropertiesofmappings.
4.Conclusions
Thispaperismainlyfocusedonsoftneutrosophicmodule.Firstly,wehaveexaminedthe generalizationandsomestructuralfeaturesofsoft-N modulesintheappropriateexamples.Then wedescribedandexemplifiedweaksoft-N modulesandstrongsoft-N modules.Weprovedthat eachstrongsoft-NG-mod isaweaksoft-NG-mod andeachweak(strong)soft-NG-mod isasoft G-mod.Finally,wedescribedthesoft-N modulehomomorphismandthesoft-N moduleisomorphism andexaminedsomepropertiesofthesoft-N modulehomomorphism.Usingtheresultsobtained, researcherscaninvestigatethehomologicalcharacteristicsofthesemodules.
AuthorContributions: Investigation,M.B.;Writing—review&editing,N.O.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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