IntroductiontoNeutroRings
AgboolaA.A.A. DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria. agboolaaaa@funaab.edu.ng
Abstract
TheobjectiveofthispaperistointroducetheconceptofNeutroRingsbyconsideringthreeNeutroAxioms (NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)andNeutroDistributivity(multiplication overaddition)).SeveralinterestingresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIdeals, NeutroQuotientRingsandNeutroRingHomomorphismsarepresented.Itisshownthatthe1stisomorphism theoremoftheclassicalringsholdsintheclassofNeutroRings.
Keywords: Neutrosophy,NeutroGroup,NeutroSubgroup,NeutroRing,NeutroSubring,NeutroIdeal,NeutroQuotientRingandNeutroRingHomomorphism.
1Introduction
Theconceptofneutrosophiclogic/setintroducedbySmarandache25 isageneralizationoffuzzylogic/setintroducedbyZadeh30 andintuitionisticfuzzylogic/setintroducedbyAtanasov.14 Inneutrosophiclogic,each propositionischaracterizedbytruthvalueintheset T ,indeterminacyvalueintheset I andfalsehoodvalue intheset F where T,I,F arestandardornonstandardofthesubsetsofthenonstandardinterval ] 0, 1+[ where 0 ≤ inf T +inf I +inf F ≤ sup T +sup I +sup F ≤ 3+.Statically, T,I,F aresubsets,but dynamically,thecomponentsof T,I,F areset-valuedvectorfunctions/operatorsdependingonmanyparameterssomeofwhichmaybehiddenorunknown.Neutrosophiclogic/sethasseveralreallifeapplicationsin sciences,engineering,technologyandsocialsciences.Theconcepthasbeenusedinmedicaldiagnosisand multipledecision-making.15,18,29 Neutrosophicsethasbeenusedandappliedinseveralareasofmathematics. Forinstanceinalgebra,neutrosophicsethasbeenusedtodevelopneutrosophicgroups,rings,vectorspaces, modules,hypergroups,hyperrings,hypervectorspaces,hypermodules,etc.1,2,4,5,7–13,16,17,27,28 Inanalysis, neutrosophicsethasbeenusedtodevelopneutrosophictopologicalspaces22–24 andmanyotherareasofmathematicalanalysis.Theconceptofneutrosophiclogic/setisnowwellknownandembracedinmanypartsof world.Manyresearcheshavebeenconductedonneutrosophiclogic/setandseveralpapershavebeenpublished inmanyinternationaljournalsbymanyneutrosophicresearchersscatteredallovertheworld.Neutrosophic SetsandSystemsandInternationalJournalofNeutrosophicSciencearepresentlytwointernationaljournals dedicatedtopublicationofresearcharticlesinneutrosophiclogic/set.
Smarandache19 recentlyintroducednewfieldsofresearchinneutrosophycalledNeutroStructuresandAntiStructuresrespectively.In,20 SmarandacheintroducedtheconceptsofNeutroAlgebrasandAntiAlgebrasand in,21 herevisitedtheconceptofNeutroAlgebrasandAntiAlgebraswherehestudiedPartialAlgebras,Universal Algebras,EffectAlgebrasandBoole’sPartialAlgebrasandheshowedthatNeutroAlgebrasaregeneralization ofPartialAlgebras.MotivatedbytheworksofSmarandachein,19–21 Agboolaetalin6 studiedNeutroAlgebras andAntiAlgebrasviz-a-viztheclassicalnumbersystems N, Z, Q, R and C.Alsomotivatedbytheworkof Smarandachein,19 Agboola3 formallyintroducedtheconceptofNeutroGroupbyconsideringthreeNeutroAxioms(NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement).In,3 AgboolastudiedNeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomorphisms.Severalinterestingresultsandexampleswerepresentedanditwasshownthatgenerally,Lagrange’s theoremand1stisomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroups.Incontinuationoftheworkstartedin,3 thepresentpaperisdevotedtothepresentationoftheconceptofNeutroRing byconsideringthreeNeutroAxioms(NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)and Doi:10.5281/zenodo.387712162
NeutroDistributivity(multiplicationoveraddition)).SeveralinterestingresultsandexamplesonNeutroRings, NeutroSubgrings,NeutroIdeals,NeutroQuotientRingsandNeutroRingHomomorphismsarepresented.Itis shownthatthe1stisomorphismtheoremoftheclassicalringsholdsintheclassofNeutroRings.
2Preliminaries
Inthissection,wewillgivesomedefinitions,examplesandresultsthatwillbeusefulinothersectionsofthe paper. Definition2.1. 21
(i) Aclassicalaxiomdefinedonanonemptysetisanaxiomthatistotallytrue(i.e.trueforallset’s elements).
(ii) ANeutroAxiom(orNeutrosophicAxiom)definedonanonemptysetisanaxiomthatistrueforsome set’selements[degreeoftruth(T)],indeterminateforotherset’selements[degreeofindeterminacy(I)], orfalsefortheotherset’selements[degreeoffalsehood(F)],where T,I,F ∈ [0, 1],with (T,I,F ) = (1, 0, 0) thatrepresentstheclassicalaxiom,and (T,I,F ) =(0, 0, 1) thatrepresentstheAntiAxiom.
(iii) AnAntiAxiomdefinedonanonemptysetisanaxiomthatisfalseforallset’selements.
Therefore,wehavetheneutrosophictriplet: < Axiom,NeutroAxiom,AntiAxiom > Definition2.2. 3 Let G beanonemptysetandlet ∗ : G × G → G beabinaryoperationon G.Thecouple (G, ∗) iscalledaNeutroGroupifthefollowingconditionsaresatisfied:
(i) ∗ isNeutroAssociativethatisthereexistsatleastonetriplet (a,b,c) ∈ G suchthat a ∗ (b ∗ c)=(a ∗ b) ∗ c (1) andthereexistsatleastonetriplet (x,y,z) ∈ G suchthat x ∗ (y ∗ z) =(x ∗ y) ∗ z. (2)
(ii) ThereexistsaNeutroNeutralelementin G thatisthereexistsatleastanelement a ∈ G thathasasingle neutralelementthatiswehave e ∈ G suchthat a ∗ e = e ∗ a = a (3) andfor b ∈ G theredoesnotexist e ∈ G suchthat b ∗ e = e ∗ b = b (4) orthereexist e1,e2 ∈ G suchthat b ∗ e1 = e1 ∗ b = b or (5) b ∗ e2 = e2 ∗ b = b with e1 = e2 (6)
(iii) ThereexistsaNeutroInverseelementthatisthereexistsanelement a ∈ G thathasaninverse b ∈ G withrespecttoaunitelement e ∈ G thatis
a ∗ b = b ∗ a = e (7)
orthereexistsatleastoneelement b ∈ G thathastwoormoreinverses c,d ∈ G withrespecttosome unitelement u ∈ G thatis
b ∗ c = c ∗ b = u (8) b ∗ d = d ∗ b = u. (9) Doi:10.5281/zenodo.387712163
Inaddition,if ∗ isNeutroCommutativethatisthereexistsatleastaduplet (a,b) ∈ G suchthat
a ∗ b = b ∗ a (10) andthereexistsatleastaduplet (c,d) ∈ G suchthat c ∗ d = d ∗ c, (11)
then (G, ∗) iscalledaNeutroCommutativeGrouporaNeutroAbelianGroup.
Ifonlycondition(i)issatisfied,then (G, ∗) iscalledaNeutroSemiGroupandifonlyconditions(i)and(ii) aresatisfied,then (G, ∗) iscalledaNeutroMonoid.
Example2.3. 3 Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet G = {a,b,c,d} beasubsetof U Let ∗ beabinaryoperationdefinedon G asshownintheCayleytablebelow:
∗ a b c d a b c d a b c d a c c d a b d d a b c a
Then (G, ∗) isaNeutroAbelianGroup.
Example2.4. 3 Let G = Z10 andlet ∗ beabinaryoperationon G definedby x ∗ y = x +2y forall x,y ∈ G where + isadditionmodulo10.Then (G, ∗) isaNeutroAbelianGroup.
Definition2.5. 3 Let (G, ∗) beaNeutroGroup.Anonemptysubset H of G iscalledaNeutroSubgroupof G if (H, ∗) isalsoaNeutroGroup.
TheonlytrivialNeutroSubgroupof G is G Example2.6. 3 Let (G, ∗) betheNeutroGroupof Example2.3 andlet H = {a,c,d}.Thecompositionsof elementsof H aregivenintheCayleytablebelow.
∗ a c d a b d a c d b d d a c a
Then, H isaNeutroSubgroupof G
.
Definition2.7. 3 Let (G, ∗) and (H, ◦) beanytwoNeutroGroups.Themapping φ : G → H iscalleda homomorphismif φ preservesthebinaryoperations ∗ and ◦ thatisifforall x,y ∈ G,wehave
φ(x ∗ y)= φ(x) ◦ φ(y) (12)
Thekernelof φ denotedby Kerφ isdefinedas
Kerφ = {x : φ(x)= eH } (13) where eH ∈ H issuchthat Nh = eH foratleastone h ∈ H.
Theimageof φ denotedby Imφ isdefinedas
Imφ(x)= {y ∈ H : y = φ(x) forsome h ∈ H} (14)
Ifinaddition φ isabijection,then φ isanisomorphismandwewrite G ∼ = H
Theorem2.8. 3 Let (G, ∗) and (H, ◦) beNeutroGroupsandlet Nx = eG suchthat eG ∗ x = x ∗ eG = x for atleastone x ∈ G andlet Ny = eH suchthat eH ∗ y = y ∗ eH = y foratleastone y ∈ H .Supposethat φ : G → H isaNeutroGrouphomomorphism.Then:
(i) φ(eG)= eH .
(ii) Kerφ isaNeutroSubgroupof G
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(iii) Imφ isaNeutroSubgroupofH.
(iv) φ isinjectiveifandonlyif Kerφ = {eg }.
Theorem2.9. 3 Let H beaNeutroSubgroupofaNeutroGroup (G, ∗).Themapping ψ : G → G/H defined by
ψ(x)= xH ∀ x ∈ G
isaNeutroGrouphomomorphismandthe Kerψ = H
Theorem2.10. 3 Let φ : G → H beaNeutroGrouphomomorpismandlet K = Kerφ.Thenthemapping ψ : G/K → Imφ definedby
ψ(xK)= φ(x) ∀ x ∈ G isaNeutroGroupepimorphismandnotanisomorphism.
3DevelopmentofNeutroRingsandtheirProperties
ThenewconceptofNeutroRingisdevelopedandstudiedinthissectionbyconsideringthreeNeutroAxioms (NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)andNeutroDistributivity(multiplication overaddition)).Severalinterestingresultsandexamplesarepresented.
Definition3.1. (a) ANeutroRing (R, +,.) isaringstructurethathasatleastoneNeutroOperationamong + and . oratleastoneNeutroAxiom.Therefore,therearemanycasesofNeutroRing,depending onthenumberofNeutroOperationsandNeutroAxioms,andwhichofthemareNeutro-Sophicated. Forthepurposesofthispaper,thefollowingdefinitionofaNeutroRingwillbeadopted:
(b) Let R beanonemptysetandlet +,. : R × R → R bebinaryoperationsofordinaryadditionand multiplicationon R.Thetriple (R, +,.) iscalledaNeutroRingifthefollowingconditionsaresatisfied:
(i) (R, +) isaNeutroAbelianGroup.
(ii) (R,.) isaNeutroSemiGroup.
(iii) isbothleftandrightNeutroDistributiveover + thatisthereexistsatleastatriplet (a,b,c) ∈ R andatleastatriplet (d,e,f ) ∈ R suchthat
a.(b + c)= a.b + a.c (15) (b + c).a = b.a + c.a (16) d.(e + f ) = d.e + d.f (17) (e + f ).d = e.d + f.d. (18)
If isNeutroCommutative,then (R, +,.) iscalledaNeutroCommutativeRing. Wewillsometimeswrite a.b = ab
Definition3.2. Let (R, +,.) beaNeutroRing.
(i) R iscalledafiniteNeutroRingoforder n ifthenumberofelementsin R is n thatis o(R)= n.Ifno such n exists,then R iscalledaninfiniteNeutroRingandwewrite o(R)= ∞
(ii) R iscalledaNeutroRingwithNeutroUnityifthereexistsamultiplicativeNeutroUnityelement u ∈ R suchthat ux = xu = x thatis Ux = u foratleastone x ∈ R.
(iii) Ifthereexistsaleastpositive n suchthat nx = e foratleastone x ∈ R where e isanadditive NeutroElementin R,then R iscalledaNeutroRingofcharacteristic n.Ifnosuch n exists,then R is calledaNeutroRingofcharacteristicNeutroZero.
(iv) Anelement x ∈ R iscalledaNeutroIdempotentelementif x2 = x
(v) Anelement x ∈ R iscalledaNeutroINilpotentelementiffortheleastpositiveinteger n,wehave xn = e where e isanadditiveNeutroNeutralelementin R
(vi) Anelement e = x ∈ R iscalledaNeutroZeroDivisorelementifthereexistsanelement e = y ∈ R such that xy = e or yx = e where e isanadditiveNeutroNeutralelementin R
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(vii) Anelement x ∈ R iscalledamultiplicativeNeutroInverseelementifthereexistsatleastone y ∈ R suchthat xy = yx = u where u isthemultiplicativeNeutroUnityelementin R
Definition3.3. Let (R, +,.) beaNeutroCommutativeRingwithNeutroUnity.Then
(i) R iscalledaNeutroIntegralDomainif R hasnoatleastoneNeutroZeroDivisorelement.
(ii) R iscalledaNeutroFieldif R hasatleastoneNeutroInverseelement.
Example3.4. Let X = {a,b,c,d} beauniverseofdiscourseandlet R = {a,b,c} beasubsetof X.Let + and . bebinaryoperationsdefinedon R asshownintheCayleytablesbelow:
+ a b c a a b b b c a c c b c a
Itisclearfromthetablethat:
a b c a a a a b a c a c a c b
c +(b + c)=(c + b)+ c = a, a +(b + c)= b, but (a + b)+ c = c = b. a + c = c + a = b, a + b = b, but b + a = c = b.
Thisshowsthat + isNeutroAssociativeandNeutroCommutative.Hence, (R, +) isacommutativeNeutroSemiGroup.
Next,let Nx and Ix representadditiveneutralelementandadditiveinverseelementrespectivelywith respecttoanyelement x ∈ R.Then
Na = a, Ia = a. Nb doesnotexist, Ib doesnotexist Nc = b, Ic = a.
Hence, (R, +) isaNeutroAbelianGroup.
Next,consider b(cb)=(bc)b = a, c(bc)= a but (cb)c = b = a.
Thisshowsthat (R,.) isNeutroAssociative.
Lastly,consider
a.(b + c)= a.b + a.c = a, b.(c + a)= c, but b.c + b.a = a = c. (b + b).b = b.b + b.b = a, (b + a).c = b, but b.c + a.c = a = b. a.b = b.a = a, b.c = a, but c.b = c = a.
Thisshowsthat isbothleftandrightNeutroDistributiveover + anditisNeutroCommutative.Hence, (R, +,.) isaNeutroCommutativeRing.Since Ua = a thatis aa = a,itfollowsthat (R, +,.) isaNeutroCommutativeRingwithNeutroUnity.
Itisobservedthat a isbothNeutroIdempotentandNeutroNilpotentelementin R.NeutroZeroDivisor elementsin R are a,b,c .Again, R isaNeutroFieldbutnotaNeutroIntegralDomain.
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Theorem3.5. EveryNeutroField R isnotnecessarilyaNeutroIntegralDomain.
Example3.6. Let X = Z10 andlet ⊕ and betwobinaryoperationson X definedby x⊕y =2x+y and x y = x +4y forall x,y ∈ X where + isadditionmodulo10.Then (X, ⊕, ) isaNeutroCommutativeRing. Toseethis:
(i) (X, ⊕) isaNeutroAbelianGroup: Let x,y,z ∈ X.Then x ⊕ (y ⊕ z)=2x +2y + z and (x ⊕ y)⊕ =4x +2y + z, equatingthesewehave 2x +2y + z =4x +2y + z ⇒ 2x =0 ∴ x =0, 5
Thus,onlythetriplets (0,x,y) and (5,x,y) canverifytheassociativityof ⊕ (degreeofassociativity = 20%)andtherefore, ⊕ isNeutroAsociative.
(ii) ExistenceofNeutroNeutralandNeutroInverseelements: Let e ∈ X suchthat x ⊕ e =2x + e = x and e ⊕ x =2e + x = x.Then 2x + e =2e + x fromwhichweobtain e = x.Butthen,only 0 ⊕ 0=0 and 5 ⊕ 5=5 in X (degreeofexistenceofneutralelement =20%).Thisshowsthat X has aNeutroNeutralelement.Itcanalsobeshownthat X hasaNeutroInverseelement.
(iii) NeutroComutativityof ⊕: Let x ⊕ y =2x + y and y ⊕ x =2y + x sothat 2x + y =2y + x fromwhichweobtain x = y.Thisshowsthatonlytheduplet (x,x) canverifycommutativityof ⊕ (degreeofcommutativity =10%)thatis, ⊕ isNeutroCommutative.Hence, (X, ⊕) isaNeutroAbelianGroup.
(iv) (X, ) isaNeutroSemiGroup: Let x,y,z ∈ X.Then x (y z)= x +4y +16z and (x y) z = x +4y +4z sothat x +4y +16z = x +4y +4z fromwhichweobtain 12z =0 sothat z =0, 5.Hence, onlythetriplets (x,y, 0) and (x,y, 5) canverifyassociativityof (degreeofassociativity =20%)and consequently, (X, ) isaNeutroSemigroup.
(v) NeutroDistributivity: Let x,y,z ∈ X.Then x (y ⊕ z)= x +8y +4z, (x y) ⊕ (x z)=3x +8y +4z sothat x +8y +4z =3x +8y +4z ⇒ 2x =0 ∴ x =0, 5.
Thisshowsthatonlythetriplets (0,y,z) and (5,y,z) canverifyleftdistributivityof over ⊕ (degreeofleftdistributivity =20%).Again, (y ⊕ z) x =4x +2y + z, (y x) ⊕ (z x)=12x +2y + z sothat 4x +2y + z =12x +2y + z ⇒ 8x =0 ∴ x =0, 5
Thisshowsthatonlythetriplets (0,y,z) and (5,y,z) canverifyrightdistributivityof over ⊕ (degreeofrightdistributivity =20%).Thus, isbothleftandrightNeutroDistributiveover ⊕.Finally, let x y = x +4y and y x = y +4x.Putting x +4y = y +4x wehave x = y showingthat onlytheduplet (x,x) canverifythecommutativityof (degreeofcommutativity =10%).Hence, isNeutroCommutativeandaccordingly, (X, ⊕, ) isaNeutroCommutativeRing.
Theorem3.7. Let (Ri, +,.),i =1, 2, ,n beafamilyofNeutroRings.Then
(i) R = n i=1 Ri isaNeutroRing.
(ii) R = n i=1 Ri isaNeutroRing.
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Proof. Obvious. Definition3.8. Let (R, +,.) beaNeutroRing.Anonemptysubset S of R iscalledaNeutroSubringof R if (S, +,.) isalsoaNeutroRing. TheonlytrivialNeutroSubringof R is R
Example3.9. Let (R, +,.) betheNeutroRingof Example3.4 andlet S = {a,b}.Thecompositionsof elementsof S aregivenintheCayleytablesbelow. + a b a a b b c a
. a b a a a b a c
Then, S isaNeutroSubringof R.Toseethis: (i) (S, +) isaNeutroAbelianGroup:
a +(a + b)=(a + a)+ b = b, b +(a + b)= a, but (b + a)+ b = c = a. Na = a, Ia = a, Nb doesnotexist, Ib doesnotexist. a + a = a, but a + b = b,b + a = c = b. Hence, (S, +) isaNeutroAbelianGroup.
(ii) (S,.) isaNeutroSemigroup:
a(ba)=(ab)a = a, b(bb)= a, but (bb)b = c = a.
Thisshowsthat (S,.) isaNeutroSemigroup.
(iii) NeutroDistributivity:
a(a + b)= aa + ab = a, b(b + a)= a, but bb + ba = b = a. (b + a)a = ba + aa = a, (a + b)a = c, but aa + ba = a = c.
ThisshowsthatbothleftandrightNeutroDistributivityhold.Accordingly, (S, +,.) isaNeutroRing. Since S isasubsetof R,itfollowsthat S isNeutroSubringof R
Theorem3.10. Let (R, +,.) beaNeutroRingandlet {Si},i =1, 2, ,n beafamilyofNeutroSubringsof R.Then
(i) S = n i=1 Si isaNeutroSubringof R.
(ii) S = n i=1 Si isaNeutroSubringof R Proof. Obvious.
Definition3.11. Let (R, +,.) beaNeutroRing.Anonemptysubset I of R iscalledaleftNeutroIdealof R if thefollowingconditionshold:
(i) I isaNeutroSubringof R
(ii) x ∈ I and r ∈ R implythatatleastone xr ∈ I forall r ∈ R
Definition3.12. Let (R, +,.) beaNeutroRing.Anonemptysubset I of R iscalledarightNeutroIdealof R ifthefollowingconditionshold:
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(i) I isaNeutroSubringof R
(ii) x ∈ I and r ∈ R implythatatleastone rx ∈ I forall r ∈ R.
Definition3.13. Let (R, +,.) beaNeutroRing.Anonemptysubset I of R iscalledaNeutroIdealof R ifthe followingconditionshold:
(i) I isaNeutroSubringof R
(ii) x ∈ I and r ∈ R implythatatleastone xr,rx ∈ I forall r ∈ R.
Example3.14. Let R = {a,b,c} betheNeutroRingof Example3.4 andlet I = S = {a,b} betheNeutroSubringof R givenin Example3.9.Considerthefollowing:
aa = a,ab = a,ac = a,ba = a,bb = c ∈ I,bc = a.
Thisshowsthat I isaleftNeutroIdealof R.Again,
aa = a,ba = a,ca = a,ab = a,bb = c ∈ I,cb = c ∈ I.
Thisalsoshowsthat I isarightNeutroIdealof R.Hence, I isaNeutroIdealof R.
Theorem3.15. Let (R, +,.) beaNeutroRingandlet {Ii},i =1, 2, ,n beafamilyofNeutroIdealsof R Then
(i) I = n i=1 Ii isaNeutroIdealof R
(ii) I = n i=1 Ii isaNeutroIdealof R
Proof. Obvious.
Definition3.16. Let (R, +,.) beaNeutroRingandlet I beaNeutroIdealof R.Theset R/I isdefinedby R/I = {x + I : x ∈ R} (19)
For x + I,y + I ∈ R/I withatleastapair (x,y) ∈ R,let ⊕ and bebinaryoperationson R/I definedas follows:
(x + I) ⊕ (y + I)=(x + y)+ I, (20) (x + I) (y + I)= xy + I. (21)
Thenitcanbeshownthatthetripple (R/I, ⊕, ) isaNeutroRingwhichwecallaNeutroQuotientRing.
Example3.17. Let R betheNeutroRingof Example3.4 andlet I beitsNeutroIdealof Example3.14.Then a + I = {a,b} = I, b + I = {a,c}, c + I = {b,c}, ∴ R/I = {a + I,b + I,c + I} = {{a,b}, {a,c}, {b,c}}
ConsidertheCayleytablesbelow:
⊕ a + I b + I c + I a + I a + I b + I b + I b + I c + I a + I c + I c + I b + I c + I a + I
a + I b + I c + I a + I a + I a + I a + I b + I a + I c + I a + I c + I a + I c + I b + I
Iteasytodeducefromthetablesthat (R/I, ⊕, ) isaNeutroRing.
Theorem3.18. Let I beaNeutroIdealoftheNeutroRing R.Then R/I isaNeutroCommutativeRingwith NeutroUnityifandonlyif R isaNeutroCommutativeRingwithNeutroUnity.
Proof. Easy.
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Definition3.19. Let (R, +,.) and (S, + ,. ) beanytwoNeutroRings.Themapping φ : R → S iscalled aNeutroRingHomomorphismif φ preservesthebinaryoperationsof R and S thatisifforatleastapair (x,y) ∈ R,wehave:
φ(x + y)= φ(x)+ φ(y), (22)
φ(x.y)= φ(x). φ(y). (23)
Thekernelof φ denotedby Kerφ isdefinedas
Kerφ = {x : φ(x)= eR} (24)
where eR ∈ R issuchthat Nr = eR foratleastone r ∈ R Theimageof φ denotedby Imφ isdefinedas
Imφ = {y ∈ S : y = φ(x) foratleastone y ∈ S} (25)
Ifinaddition φ isaNeutroBijection,then φ iscalledaNeutroRingIsomorphismandwewrite R ∼ = S NeutroRingEpimorphism,NeutroRingMonomorphism,NeutroRingEndomorphismandNeutroRingAutomorphismaredefinedsimilarly.
Example3.20. Let R betheNeutroRingof Example3.4 andlet φ : R × R → R beamappingdefinedby
φ((x,y))= a if x = y, d if x = y.
Itcanbeshownthat φ isaNeutroRingHomomorphism.The Kerφ = {(a,a), (b,b), (c,c)} whichisaNeutroSubgRingof R × R ascanbeseenintheCayleytablesbelow.
+ (a,a) (b,b) (c,c) (a,a) (a,a) (b,b) (b,b) (b,b) (c,c) (a,a) (c,c) (c,c) (b,b) (c,c) (a,a)
Imφ = {a,d} ⊆ R
(a,a) (b,b) (c,c) (a,a) (a,a) (a,a) (a,a) (b,b) (a,a) (c,c) (a,a) (c,c) (a,a) (c,c) (b,b)
Theorem3.21. Let R and S betwoNeutroRings.Let Nx = eR foratleastone x ∈ R andlet Ny = eS forat leastone y ∈ S.Supposethat φ : R → S isaNeutroRingHomomorphism.Then:
(i) φ(eR) isnotnecessarilyequals eS
(ii) Kerφ isaNeutroSubringof R
(iii) Imφ isnotnecessarilyaNeutroSubringof S
(iv) φ isNeutroInjectiveifandonlyif Kerφ = {eR} foratleastone eR ∈ R
Example3.22. Let R = {a,b,c} betheNeutroRingof Example3.4 andlet I = {a,b} betheNeutroIdealof R givenin Example3.14.Let φ : R → R/I beamappingdefinedby φ(x)= x + I foratleastone x ∈ R Then φ(a)= a + I = {a,b} = I,φ(b)= b + I = {a,c} and φ(c)= c + I = {b,c} fromwhichweobtain that φ isaNeutroRingHomomorphism.
Kerφ = {x ∈ R : φ(x)= eR/I } = {x ∈ R : x + I = eR/I = a + I} = I.
Theorem3.23. Let I beaNeutroIdealofaNeutroRing R.Thenthemapping ψ : R → R/I definedby
ψ(x)= x + I foratleastone x ∈ R
isaNeutroRingEpimomorphismandthe Kerψ = I .
Theorem3.24. Let φ : R → S beaNeutroRingHomomorpismandlet K = Kerφ.Thenthemapping ψ : R/K → Imφ definedby
ψ(x + K)= φ(x) foratleastone x ∈ R
isaNeutroRingIsomorphism.
Doi:10.5281/zenodo.387712170
Proof. Let x + K,y + K ∈ R/K withatleastapair (x,y) ∈ R.Then
ψ((x + K) ⊕ (y + K))= ψ((x + y)+ K)
= φ(x + y)
= φ(x)+ φ(y)
= ψ(x + K) ⊕ ψ(y + K)
ψ((x + K) (y + K))= ψ((xy)+ K)
= φ(xy)
= φ(x)φ(y)
= ψ(x + K) ψ(y + K)
Kerψ = {x + K ∈ R/K : ψ(x + K)= eφ(x)}
= {x + K ∈ R/K : φ(x)= eφ(x)}
= {eR/K }.
Thisshowsthat ψ isaNeutroBijectiveHomomorphismandthereforeitisaNeutroRingIsomorphismthatis R/K ∼ = Imφ whichisthesameaswhatisobtainableintheclassicalrings.
Theorem3.25. NeutroRingIsomorphismofNeutroRingsisanequivalencerelation.
Proof. Theproofisthesameastheclassicalrings.
4Conclusion
WehaveforthefirsttimeintroducedinthispapertheconceptofNeutroRingsbyconsideringthreeNeutroAxioms(NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)andNeutroDistributivity(multiplicationoveraddition)).SeveralinterestingresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIdeals,NeutroQuotientRingsandNeutroRingHomomorphismsarepresented.Itisshownthatthe1stisomorphism theoremoftheclassicalringsholdsintheclassofNeutroRings.MoreadvancedpropertiesofNeutroRings willbepresentedinourfuturepapers.OtherNeutroAlgebraicStructuressuchasNeutroModules,NeutroVectorSpacesetcareopenedtobedevelopedandstudiedbyotherNeutrosophicresearchers.
5Appreciation
Theauthorisverygratefultoallanonymousreviewersfortheirvaluableobservations,commentsandsuggestionswhichhavecontributedimmenselytothequalityofthepaper.
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