Table Of ContentInternationalJournalofNeutrosophicScience(IJNS) Vol. 7,No. 2,PP.62-73,2020
Introduction to NeutroRings
AgboolaA.A.A.
DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria.
[email protected]
Abstract
The objective of this paper is to introduce the concept of NeutroRings by considering three NeutroAxioms
(NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication
over addition)). Several interesting results and examples on NeutroRings, NeutroSubgrings, NeutroIdeals,
NeutroQuotientRings and NeutroRingHomomorphisms are presented. It is shown that the 1st isomorphism
theoremoftheclassicalringsholdsintheclassofNeutroRings.
Keywords: Neutrosophy,NeutroGroup,NeutroSubgroup,NeutroRing,NeutroSubring,NeutroIdeal,Neutro-
QuotientRingandNeutroRingHomomorphism.
1 Introduction
Theconceptofneutrosophiclogic/setintroducedbySmarandache25 isageneralizationoffuzzylogic/setin-
troducedbyZadeh30 andintuitionisticfuzzylogic/setintroducedbyAtanasov.14 Inneutrosophiclogic,each
propositionischaracterizedbytruthvalueinthesetT,indeterminacyvalueinthesetI andfalsehoodvalue
in the set F where T,I,F are standard or nonstandard of the subsets of the nonstandard interval ]−0,1+[
where −0 ≤ infT +infI +infF ≤ supT +supI +supF ≤ 3+. Statically, T,I,F are subsets, but
dynamically,thecomponentsofT,I,F areset-valuedvectorfunctions/operatorsdependingonmanyparam-
eters some of which may be hidden or unknown. Neutrosophic logic/set has several real life applications in
sciences, engineering, technology and social sciences. The concept has been used in medical diagnosis and
multipledecision-making.15,18,29 Neutrosophicsethasbeenusedandappliedinseveralareasofmathematics.
Forinstanceinalgebra,neutrosophicsethasbeenusedtodevelopneutrosophicgroups,rings,vectorspaces,
modules, hypergroups, hyperrings, hypervector spaces, hypermodules, etc.1,2,4,5,7–13,16,17,27,28 In analysis,
neutrosophicsethasbeenusedtodevelopneutrosophictopologicalspaces22–24andmanyotherareasofmath-
ematical analysis. The concept of neutrosophic logic/set is now well known and embraced in many parts of
world.Manyresearcheshavebeenconductedonneutrosophiclogic/setandseveralpapershavebeenpublished
in many international journals by many neutrosophic researchers scattered all over the world. Neutrosophic
Sets and Systemsand International Journal of Neutrosophic Scienceare presently two international journals
dedicatedtopublicationofresearcharticlesinneutrosophiclogic/set.
Smarandache19recentlyintroducednewfieldsofresearchinneutrosophycalledNeutroStructuresandAn-
tiStructuresrespectively. In,20SmarandacheintroducedtheconceptsofNeutroAlgebrasandAntiAlgebrasand
in,21herevisitedtheconceptofNeutroAlgebrasandAntiAlgebraswherehestudiedPartialAlgebras,Universal
Algebras,EffectAlgebrasandBoole’sPartialAlgebrasandheshowedthatNeutroAlgebrasaregeneralization
ofPartialAlgebras.MotivatedbytheworksofSmarandachein,19–21Agboolaetalin6studiedNeutroAlgebras
and AntiAlgebras viz-a-viz the classical number systems N, Z, Q, R and C. Also motivated by the work of
Smarandachein,19Agboola3formallyintroducedtheconceptofNeutroGroupbyconsideringthreeNeutroAx-
ioms(NeutroAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement). In,3
AgboolastudiedNeutroSubgroups,NeutroCyclicGroups,NeutroQuotientGroupsandNeutroGroupHomomor-
phisms. Severalinterestingresultsandexampleswerepresentedanditwasshownthatgenerally,Lagrange’s
theoremand1stisomorphismtheoremoftheclassicalgroupsdonotholdintheclassofNeutroGroups.Incon-
tinuationoftheworkstartedin,3thepresentpaperisdevotedtothepresentationoftheconceptofNeutroRing
by considering three NeutroAxioms (NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and
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NeutroDistributivity(multiplicationoveraddition)). SeveralinterestingresultsandexamplesonNeutroRings,
NeutroSubgrings, NeutroIdeals, NeutroQuotientRings and NeutroRingHomomorphisms are presented. It is
shownthatthe1stisomorphismtheoremoftheclassicalringsholdsintheclassofNeutroRings.
2 Preliminaries
Inthissection,wewillgivesomedefinitions,examplesandresultsthatwillbeusefulinothersectionsofthe
paper.
Definition2.1. 21
(i) A classical axiom defined on a nonempty set is an axiom that is totally true (i.e. true for all set’s
elements).
(ii) ANeutroAxiom(orNeutrosophicAxiom)definedonanonemptysetisanaxiomthatistrueforsome
set’selements[degreeoftruth(T)],indeterminateforotherset’selements[degreeofindeterminacy(I)],
orfalsefortheotherset’selements[degreeoffalsehood(F)],whereT,I,F ∈ [0,1],with(T,I,F) (cid:54)=
(1,0,0)thatrepresentstheclassicalaxiom,and(T,I,F)(cid:54)=(0,0,1)thatrepresentstheAntiAxiom.
(iii) AnAntiAxiomdefinedonanonemptysetisanaxiomthatisfalseforallset’selements.
Therefore,wehavetheneutrosophictriplet: <Axiom,NeutroAxiom,AntiAxiom>.
Definition2.2. 3 LetGbeanonemptysetandlet∗ : G×G → GbeabinaryoperationonG. Thecouple
(G,∗)iscalledaNeutroGroupifthefollowingconditionsaresatisfied:
(i) ∗isNeutroAssociativethatisthereexistsatleastonetriplet(a,b,c)∈Gsuchthat
a∗(b∗c)=(a∗b)∗c (1)
andthereexistsatleastonetriplet(x,y,z)∈Gsuchthat
x∗(y∗z)(cid:54)=(x∗y)∗z. (2)
(ii) ThereexistsaNeutroNeutralelementinGthatisthereexistsatleastanelementa∈Gthathasasingle
neutralelementthatiswehavee∈Gsuchthat
a∗e=e∗a=a (3)
andforb∈Gtheredoesnotexiste∈Gsuchthat
b∗e=e∗b=b (4)
orthereexiste ,e ∈Gsuchthat
1 2
b∗e = e ∗b=b or (5)
1 1
b∗e = e ∗b=b with e (cid:54)=e (6)
2 2 1 2
(iii) There exists a NeutroInverse element that is there exists an element a ∈ G that has an inverse b ∈ G
withrespecttoaunitelemente∈Gthatis
a∗b=b∗a=e (7)
orthereexistsatleastoneelementb ∈ Gthathastwoormoreinversesc,d ∈ Gwithrespecttosome
unitelementu∈Gthatis
b∗c = c∗b=u (8)
b∗d = d∗b=u. (9)
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Inaddition,if∗isNeutroCommutativethatisthereexistsatleastaduplet(a,b)∈Gsuchthat
a∗b=b∗a (10)
andthereexistsatleastaduplet(c,d)∈Gsuchthat
c∗d(cid:54)=d∗c, (11)
then(G,∗)iscalledaNeutroCommutativeGrouporaNeutroAbelianGroup.
Ifonlycondition(i)issatisfied,then(G,∗)iscalledaNeutroSemiGroupandifonlyconditions(i)and(ii)
aresatisfied,then(G,∗)iscalledaNeutroMonoid.
Example2.3. 3 LetU={a,b,c,d,e,f}beauniverseofdiscourseandletG={a,b,c,d}beasubsetofU.
Let∗beabinaryoperationdefinedonGasshownintheCayleytablebelow:
∗ 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. 3LetG=Z andlet∗beabinaryoperationonGdefinedbyx∗y =x+2yforallx,y ∈G
10
where(cid:48)(cid:48)+(cid:48)(cid:48)isadditionmodulo10. Then(G,∗)isaNeutroAbelianGroup.
Definition2.5. 3 Let(G,∗)beaNeutroGroup. AnonemptysubsetH ofGiscalledaNeutroSubgroupofG
if(H,∗)isalsoaNeutroGroup.
TheonlytrivialNeutroSubgroupofGisG.
Example2.6. 3 Let(G,∗)betheNeutroGroupofExample2.3andletH = {a,c,d}. Thecompositionsof
elementsofH aregivenintheCayleytablebelow.
∗ a c d
a b d a
.
c d b d
d a c a
Then,H isaNeutroSubgroupofG.
Definition 2.7. 3 Let (G,∗) and (H,◦) be any two NeutroGroups. The mapping φ : G → H is called a
homomorphismifφpreservesthebinaryoperations∗and◦thatisifforallx,y ∈G,wehave
φ(x∗y)=φ(x)◦φ(y). (12)
ThekernelofφdenotedbyKerφisdefinedas
Kerφ={x:φ(x)=e } (13)
H
wheree ∈H issuchthatN =e foratleastoneh∈H.
H h H
TheimageofφdenotedbyImφisdefinedas
Imφ(x)={y ∈H :y =φ(x) forsome h∈H}. (14)
Ifinadditionφisabijection,thenφisanisomorphismandwewriteG∼=H.
Theorem2.8. 3 Let(G,∗)and(H,◦)beNeutroGroupsandletN =e suchthate ∗x=x∗e =xfor
x G G G
atleastonex ∈ GandletN = e suchthate ∗y = y∗e = y foratleastoney ∈ H. Supposethat
y H H H
φ:G→H isaNeutroGrouphomomorphism. Then:
(i) φ(e )=e .
G H
(ii) KerφisaNeutroSubgroupofG.
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(iii) ImφisaNeutroSubgroupofH.
(iv) φisinjectiveifandonlyifKerφ={e }.
g
Theorem2.9. 3 LetH beaNeutroSubgroupofaNeutroGroup(G,∗). Themappingψ : G → G/H defined
by
ψ(x)=xH ∀x∈G
isaNeutroGrouphomomorphismandtheKerψ (cid:54)=H.
Theorem2.10. 3 Letφ : G → H beaNeutroGrouphomomorpismandletK = Kerφ. Thenthemapping
ψ :G/K →Imφdefinedby
ψ(xK)=φ(x) ∀x∈G
isaNeutroGroupepimorphismandnotanisomorphism.
3 Development of NeutroRings and their Properties
ThenewconceptofNeutroRingisdevelopedandstudiedinthissectionbyconsideringthreeNeutroAxioms
(NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication
overaddition)). Severalinterestingresultsandexamplesarepresented.
Definition3.1. (a) ANeutroRing(R,+,.)isaringstructurethathasatleastoneNeutroOperationamong
(cid:48)(cid:48)+(cid:48)(cid:48) and(cid:48)(cid:48).(cid:48)(cid:48) oratleastoneNeutroAxiom. Therefore,therearemanycasesofNeutroRing,depending
onthenumberofNeutroOperationsandNeutroAxioms,andwhichofthemareNeutro-Sophicated.
Forthepurposesofthispaper,thefollowingdefinitionofaNeutroRingwillbeadopted:
(b) Let R be a nonempty set and let +,. : R ×R → R be binary operations of ordinary addition and
multiplicationonR. Thetriple(R,+,.)iscalledaNeutroRingifthefollowingconditionsaresatisfied:
(i) (R,+)isaNeutroAbelianGroup.
(ii) (R,.)isaNeutroSemiGroup.
(iii) (cid:48)(cid:48).(cid:48)(cid:48)isbothleftandrightNeutroDistributiveover(cid:48)(cid:48)+(cid:48)(cid:48)thatisthereexistsatleastatriplet(a,b,c)∈
Randatleastatriplet(d,e,f)∈Rsuchthat
a.(b+c) = a.b+a.c (15)
(b+c).a = b.a+c.a (16)
d.(e+f) (cid:54)= d.e+d.f (17)
(e+f).d (cid:54)= e.d+f.d. (18)
If(cid:48)(cid:48).(cid:48)(cid:48)isNeutroCommutative,then(R,+,.)iscalledaNeutroCommutativeRing.
Wewillsometimeswritea.b=ab.
Definition3.2. Let(R,+,.)beaNeutroRing.
(i) RiscalledafiniteNeutroRingofordernifthenumberofelementsinRisnthatiso(R) = n. Ifno
suchnexists,thenRiscalledaninfiniteNeutroRingandwewriteo(R)=∞.
(ii) RiscalledaNeutroRingwithNeutroUnityifthereexistsamultiplicativeNeutroUnityelementu ∈ R
suchthatux=xu=xthatisU =uforatleastonex∈R.
x
(iii) If there exists a least positive n such that nx = e for at least one x ∈ R where e is an additive
NeutroElement in R, then R is called a NeutroRing of characteristic n. If no such n exists, then R is
calledaNeutroRingofcharacteristicNeutroZero.
(iv) Anelementx∈RiscalledaNeutroIdempotentelementifx2 =x.
(v) An element x ∈ R is called a NeutroINilpotent element if for the least positive integer n, we have
xn =ewhereeisanadditiveNeutroNeutralelementinR.
(vi) Anelemente(cid:54)=x∈RiscalledaNeutroZeroDivisorelementifthereexistsanelemente(cid:54)=y ∈Rsuch
thatxy =eoryx=ewhereeisanadditiveNeutroNeutralelementinR.
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(vii) An element x ∈ R is called a multiplicative NeutroInverse element if there exists at least one y ∈ R
suchthatxy =yx=uwhereuisthemultiplicativeNeutroUnityelementinR.
Definition3.3. Let(R,+,.)beaNeutroCommutativeRingwithNeutroUnity. Then
(i) RiscalledaNeutroIntegralDomainifRhasnoatleastoneNeutroZeroDivisorelement.
(ii) RiscalledaNeutroFieldifRhasatleastoneNeutroInverseelement.
Example3.4. LetX={a,b,c,d}beauniverseofdiscourseandletR={a,b,c}beasubsetofX. Let(cid:48)(cid:48)+(cid:48)(cid:48)
and(cid:48)(cid:48).(cid:48)(cid:48)bebinaryoperationsdefinedonRasshownintheCayleytablesbelow:
+ a b c . a b c
a a b b a a a a
b c a c b a c a
c b c a c a c b
Itisclearfromthetablethat:
c+(b+c) = (c+b)+c=a,
a+(b+c) = b, but (a+b)+c=c(cid:54)=b.
a+c = c+a=b,
a+b = b, but b+a=c(cid:54)=b.
This shows that (cid:48)(cid:48)+(cid:48)(cid:48) is NeutroAssociative and NeutroCommutative. Hence, (R,+) is a commutative Neu-
troSemiGroup.
Next, let N and I represent additive neutral element and additive inverse element respectively with
x x
respecttoanyelementx∈R. Then
N = a,
a
I = a.
a
N doesnotexist,
b
I doesnotexist.
b
N = b,
c
I = a.
c
Hence,(R,+)isaNeutroAbelianGroup.
Next,consider
b(cb) = (bc)b=a,
c(bc) = a but (cb)c=b(cid:54)=a.
Thisshowsthat(R,.)isNeutroAssociative.
Lastly,consider
a.(b+c) = a.b+a.c=a,
b.(c+a) = c, but b.c+b.a=a(cid:54)=c.
(b+b).b = b.b+b.b=a,
(b+a).c = b, but b.c+a.c=a(cid:54)=b.
a.b = b.a=a,
b.c = a, but c.b=c(cid:54)=a.
Thisshowsthat(cid:48)(cid:48).(cid:48)(cid:48) isbothleftandrightNeutroDistributiveover(cid:48)(cid:48)+(cid:48)(cid:48) anditisNeutroCommutative. Hence,
(R,+,.)isaNeutroCommutativeRing. SinceU =athatisaa=a,itfollowsthat(R,+,.)isaNeutroCom-
a
mutativeRingwithNeutroUnity.
Itisobservedthat(cid:48)(cid:48)a(cid:48)(cid:48) isbothNeutroIdempotentandNeutroNilpotentelementinR. NeutroZeroDivisor
elementsinRare(cid:48)(cid:48)a,b,c(cid:48)(cid:48). Again,RisaNeutroFieldbutnotaNeutroIntegralDomain.
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Theorem3.5. EveryNeutroFieldRisnotnecessarilyaNeutroIntegralDomain.
Example3.6. LetX =Z andlet⊕and(cid:12)betwobinaryoperationsonXdefinedbyx⊕y =2x+yandx(cid:12)
10
y =x+4yforallx,y ∈X where(cid:48)(cid:48)+(cid:48)(cid:48)isadditionmodulo10. Then(X,⊕,(cid:12))isaNeutroCommutativeRing.
Toseethis:
(i) (X,⊕)isaNeutroAbelianGroup: Letx,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: Lete ∈ X suchthatx⊕e = 2x+e = x
and e⊕x = 2e+x = x. Then 2x+e = 2e+x from which we obtain e = x. But then, only
0⊕0=0and5⊕5=5inX (degreeofexistenceofneutralelement=20%). ThisshowsthatX has
aNeutroNeutralelement. ItcanalsobeshownthatX hasaNeutroInverseelement.
(iii) NeutroComutativity of ⊕: Let x ⊕ y = 2x + y and y ⊕ x = 2y + x so that 2x + y = 2y + x
from which we obtain x = y. This shows that only the duplet (x,x) can verify commutativity of ⊕
(degreeofcommutativity=10%)thatis,⊕isNeutroCommutative. Hence,(X,⊕)isaNeutroAbelian-
Group.
(iv) (X,(cid:12))isaNeutroSemiGroup: Letx,y,z ∈X. Then
x(cid:12)(y(cid:12)z)=x+4y+16z and (x(cid:12)y)(cid:12)z =x+4y+4z
so that x + 4y + 16z = x + 4y + 4z from which we obtain 12z = 0 so that z = 0,5. Hence,
onlythetriplets(x,y,0)and(x,y,5)canverifyassociativityof(cid:12)(degreeofassociativity=20%)and
consequently,(X,(cid:12))isaNeutroSemigroup.
(v) NeutroDistributivity: Letx,y,z ∈X. Then
x(cid:12)(y⊕z) = x+8y+4z,(x(cid:12)y)⊕(x(cid:12)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(cid:12)over⊕
(degreeofleftdistributivity=20%). Again,
(y⊕z)(cid:12)x = 4x+2y+z,(y(cid:12)x)⊕(z(cid:12)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(cid:12)over⊕
(degreeofrightdistributivity=20%).Thus,(cid:12)isbothleftandrightNeutroDistributiveover⊕.Finally,
let x(cid:12)y = x+4y and y (cid:12)x = y +4x. Putting x+4y = y +4x we have x = y showing that
onlytheduplet(x,x)canverifythecommutativityof(cid:12)(degreeofcommutativity = 10%). Hence,(cid:12)
isNeutroCommutativeandaccordingly,(X,⊕,(cid:12))isaNeutroCommutativeRing.
Theorem3.7. Let(R ,+,.),i=1,2,··· ,nbeafamilyofNeutroRings. Then
i
(i) R=(cid:84)n R isaNeutroRing.
i=1 i
(ii) R=(cid:81)n R isaNeutroRing.
i=1 i
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Proof. Obvious.
Definition3.8. Let(R,+,.)beaNeutroRing. AnonemptysubsetS ofRiscalledaNeutroSubringofRif
(S,+,.)isalsoaNeutroRing.
TheonlytrivialNeutroSubringofRisR.
Example 3.9. Let (R,+,.) be the NeutroRing of Example 3.4 and let S = {a,b}. The compositions of
elementsofS aregivenintheCayleytablesbelow.
+ a b . a b
a a b a a a
b c a b a c
Then,S isaNeutroSubringofR. Toseethis:
(i) (S,+)isaNeutroAbelianGroup:
a+(a+b) = (a+a)+b=b,
b+(a+b) = a, but (b+a)+b=c(cid:54)=a.
N = a,
a
I = a,
a
N doesnotexist,
b
I doesnotexist.
b
a+a = a, but a+b=b,b+a=c(cid:54)=b.
Hence,(S,+)isaNeutroAbelianGroup.
(ii) (S,.)isaNeutroSemigroup:
a(ba) = (ab)a=a,
b(bb) = a, but (bb)b=c(cid:54)=a.
Thisshowsthat(S,.)isaNeutroSemigroup.
(iii) NeutroDistributivity:
a(a+b) = aa+ab=a,
b(b+a) = a, but bb+ba=b(cid:54)=a.
(b+a)a = ba+aa=a,
(a+b)a = c, but aa+ba=a(cid:54)=c.
This shows that both left and right NeutroDistributivity hold. Accordingly, (S,+,.) is a NeutroRing.
SinceS isasubsetofR,itfollowsthatS isNeutroSubringofR.
Theorem3.10. Let(R,+,.)beaNeutroRingandlet{S },i = 1,2,··· ,nbeafamilyofNeutroSubringsof
i
R. Then
(i) S =(cid:84)n S isaNeutroSubringofR.
i=1 i
(ii) S =(cid:81)n S isaNeutroSubringofR.
i=1 i
Proof. Obvious.
Definition3.11. Let(R,+,.)beaNeutroRing. AnonemptysubsetI ofRiscalledaleftNeutroIdealofRif
thefollowingconditionshold:
(i) I isaNeutroSubringofR.
(ii) x∈I andr ∈Rimplythatatleastonexr ∈I forallr ∈R.
Definition3.12. Let(R,+,.)beaNeutroRing. AnonemptysubsetI ofRiscalledarightNeutroIdealofR
ifthefollowingconditionshold:
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(i) I isaNeutroSubringofR.
(ii) x∈I andr ∈Rimplythatatleastonerx∈I forallr ∈R.
Definition3.13. Let(R,+,.)beaNeutroRing. AnonemptysubsetI ofRiscalledaNeutroIdealofRifthe
followingconditionshold:
(i) I isaNeutroSubringofR.
(ii) x∈I andr ∈Rimplythatatleastonexr,rx∈I forallr ∈R.
Example3.14. LetR = {a,b,c}betheNeutroRingofExample3.4andletI = S = {a,b}betheNeutro-
SubringofRgiveninExample3.9. Considerthefollowing:
aa=a,ab=a,ac=a,ba=a,bb=c(cid:54)∈I,bc=a.
ThisshowsthatI isaleftNeutroIdealofR. Again,
aa=a,ba=a,ca=a,ab=a,bb=c(cid:54)∈I,cb=c(cid:54)∈I.
ThisalsoshowsthatI isarightNeutroIdealofR. Hence,I isaNeutroIdealofR.
Theorem3.15. Let(R,+,.)beaNeutroRingandlet{I },i = 1,2,··· ,nbeafamilyofNeutroIdealsofR.
i
Then
(i) I =(cid:84)n I isaNeutroIdealofR.
i=1 i
(ii) I =(cid:80)n I isaNeutroIdealofR.
i=1 i
Proof. Obvious.
Definition3.16. Let(R,+,.)beaNeutroRingandletI beaNeutroIdealofR. ThesetR/I isdefinedby
R/I ={x+I :x∈R}. (19)
Forx+I,y+I ∈ R/I withatleastapair(x,y) ∈ R,let⊕and(cid:12)bebinaryoperationsonR/I definedas
follows:
(x+I)⊕(y+I) = (x+y)+I, (20)
(x+I)(cid:12)(y+I) = xy+I. (21)
Thenitcanbeshownthatthetripple(R/I,⊕,(cid:12))isaNeutroRingwhichwecallaNeutroQuotientRing.
Example3.17. LetRbetheNeutroRingofExample3.4andletI beitsNeutroIdealofExample3.14. 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 (cid:12) a+I b+I c+I
a+I a+I b+I b+I a+I a+I a+I a+I
b+I c+I a+I c+I b+I a+I c+I a+I
c+I b+I c+I a+I c+I a+I c+I b+I
Iteasytodeducefromthetablesthat(R/I,⊕,(cid:12))isaNeutroRing.
Theorem 3.18. Let I be a NeutroIdeal of the NeutroRing R. Then R/I is a NeutroCommutativeRing with
NeutroUnityifandonlyifRisaNeutroCommutativeRingwithNeutroUnity.
Proof. Easy.
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Definition 3.19. Let (R,+,.) and (S,+(cid:48),.(cid:48)) be any two NeutroRings. The mapping φ : R → S is called
a NeutroRingHomomorphism if φ preserves the binary operations of R and S that is if for at least a pair
(x,y)∈R,wehave:
φ(x+y) = φ(x)+(cid:48)φ(y), (22)
φ(x.y) = φ(x).(cid:48)φ(y). (23)
ThekernelofφdenotedbyKerφisdefinedas
Kerφ={x:φ(x)=e } (24)
R
wheree ∈RissuchthatN =e foratleastoner ∈R.
R r R
TheimageofφdenotedbyImφisdefinedas
Imφ={y ∈S :y =φ(x) foratleastone y ∈S}. (25)
If in addition φ is a NeutroBijection, then φ is called a NeutroRingIsomorphism and we write R ∼= S.
NeutroRingEpimorphism,NeutroRingMonomorphism,NeutroRingEndomorphismandNeutroRingAutomor-
phismaredefinedsimilarly.
Example3.20. LetRbetheNeutroRingofExample3.4andletφ:R×R→Rbeamappingdefinedby
(cid:26)
a if x=y,
φ((x,y))=
d if x(cid:54)=y.
ItcanbeshownthatφisaNeutroRingHomomorphism. TheKerφ = {(a,a),(b,b),(c,c)}whichisaNeu-
troSubgRingofR×RascanbeseenintheCayleytablesbelow.
+ (a,a) (b,b) (c,c) . (a,a) (b,b) (c,c)
(a,a) (a,a) (b,b) (b,b) (a,a) (a,a) (a,a) (a,a)
(b,b) (c,c) (a,a) (c,c) (b,b) (a,a) (c,c) (a,a)
(c,c) (b,b) (c,c) (a,a) (c,c) (a,a) (c,c) (b,b)
Imφ={a,d}(cid:54)⊆R.
Theorem3.21. LetRandSbetwoNeutroRings. LetN =e foratleastonex∈RandletN =e forat
x R y S
leastoney ∈S. Supposethatφ:R→S isaNeutroRingHomomorphism. Then:
(i) φ(e )isnotnecessarilyequalse .
R S
(ii) KerφisaNeutroSubringofR.
(iii) ImφisnotnecessarilyaNeutroSubringofS.
(iv) φisNeutroInjectiveifandonlyifKerφ={e }foratleastonee ∈R.
R R
Example3.22. LetR={a,b,c}betheNeutroRingofExample3.4andletI ={a,b}betheNeutroIdealof
RgiveninExample3.14. Letφ : R → R/I beamappingdefinedbyφ(x) = x+I foratleastonex ∈ R.
Thenφ(a) = a+I = {a,b} = I,φ(b) = b+I = {a,c}andφ(c) = c+I = {b,c}fromwhichweobtain
thatφisaNeutroRingHomomorphism.
Kerφ={x∈R:φ(x)=e }={x∈R:x+I =e =a+I}=I.
R/I R/I
Theorem3.23. LetI beaNeutroIdealofaNeutroRingR. Thenthemappingψ :R→R/I definedby
ψ(x)=x+I foratleastone x∈R
isaNeutroRingEpimomorphismandtheKerψ =I.
Theorem 3.24. Let φ : R → S be a NeutroRingHomomorpism and let K = Kerφ. Then the mapping
ψ :R/K →Imφdefinedby
ψ(x+K)=φ(x) foratleastone x∈R
isaNeutroRingIsomorphism.
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Proof. Letx+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)(cid:12)(y+K)) = ψ((xy)+K)
= φ(xy)
= φ(x)φ(y)
= ψ(x+K)(cid:12)ψ(y+K).
Kerψ = {x+K ∈R/K :ψ(x+K)=e }
φ(x)
= {x+K ∈R/K :φ(x)=e }
φ(x)
= {e }.
R/K
This shows that ψ is a NeutroBijectiveHomomorphism and therefore it is a NeutroRingIsomorphism that is
R/K ∼=Imφwhichisthesameaswhatisobtainableintheclassicalrings.
Theorem3.25. NeutroRingIsomorphismofNeutroRingsisanequivalencerelation.
Proof. Theproofisthesameastheclassicalrings.
4 Conclusion
WehaveforthefirsttimeintroducedinthispapertheconceptofNeutroRingsbyconsideringthreeNeutroAx-
ioms (NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multipli-
cationoveraddition)). SeveralinterestingresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIde-
als,NeutroQuotientRingsandNeutroRingHomomorphismsarepresented.Itisshownthatthe1stisomorphism
theorem of the classical rings holds in the class of NeutroRings. More advanced properties of NeutroRings
willbepresentedinourfuturepapers. OtherNeutroAlgebraicStructuressuchasNeutroModules,NeutroVec-
torSpacesetcareopenedtobedevelopedandstudiedbyotherNeutrosophicresearchers.
5 Appreciation
Theauthorisverygratefultoallanonymousreviewersfortheirvaluableobservations,commentsandsugges-
tionswhichhavecontributedimmenselytothequalityofthepaper.
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