OnFiniteandInfiniteNeutroRingsofType-NR[8,9]
A.A.A.Agboola
DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria. agboolaaaa@funaab.edu.ng Incommemorationofthe60thbirthdayoftheauthor
Abstract
NeutroRingsarealternativestotheclassicalringsandtheyareofdifferenttypes.NeutroRingsinsomecases exhibitdifferentalgebraicproperties,andinsomecasestheyexhibitalgebraicpropertiessimilartotheclassicalrings.TheobjectiveofthispaperistorevisittheconceptofNeutroRingsandstudyfiniteandinfinite NeutroRingsoftype-NR[8,9].InNeutroRingsoftype-NR[8,9],theleftandrightdistributiveaxiomsaretakingtobeeitherpartiallytrueorpartiallyfalseforsomeelements;whileallotherclassicallawsandaxiomsare takingtobetotallytrueforalltheelements.SeveralexamplesandpropertiesofNeutroRingsoftype-NR[8,9] arepresented.NeutroSubrings,NeutroIdeals,NeutroQuotientRingsandNeutroRingHomomorphismsofthe NeutroRingsoftype-NR[8,9]arestudiedwithseveralinterestingexamplesandtheirbasicpropertiesarepresented.ItisshownthatinNeutroRingsoftype-NR[8,9],thefundamentaltheoremofhomomorphismsofthe classicalringsholds.
Keywords: NeutroRing,AntiRing,NeutroSubring,NeutroIdeal,NeutroQuotientRing,NeutroRingHomomorphism.
1IntroductionandPreliminaries
Intheclassicalrings (R, +,.),additionandmultiplicationclosurelawsare100%trueforalltheelements of R.Also,associativeanddistributiveaxiomsover R are100%trueforalltheelementsof R.There arenoprovisionsintheclassicalring R tohaveadditionandmultiplicationlawstobeeitherpartiallytrueor partiallyindeterminateorpartiallyfalsefortheelementsof R.Also,therearenoprovisionsforassociativeand distributiveaxiomsover R tobeeitherpartiallytrueorpartiallyindeterminateorpartiallyfalsefortheelements of R.Lackoftheseprovisionsintheclassicalringspossesproblemsbecausesuchringscannotbeusedto modelthereallifesituationsaccurately.TheseproblemswereaddressedbySmarandachein[10]byintroducing theconceptsofNeutroAlgebraicStructuresandAntiAlgebraicStructures.Smarandachefurtherstudiedthese newconceptsin[9]and[8]respectively.Withthesenewconcepts,alotofresearchactivitieshavebegunwith somepapersalreadypublished.Forinstancein[7],RezaeiandSmarandachestudiedNeutro-BE-algebrasand Anti-BE-algebrasandtheyshowedthatanyclassicalalgebra S with n operations(lawsandaxioms)where n ≥ 1 willhave (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.In[3],Agboolaetal.studiedNeutroAlgebras andAntiAlgebrasviz-a-viztheclassicalnumbersystems,in[4],AgboolastudiedNeutroGroupsandin[5],he studiedNeutroRings.Alsoin[2],AgboolarevisitedNeutroGroupsandin[1],hestudiedAntiGroups.Inthe presentpaper,theconceptofNeutroRingsintroducedin[5]isrevisited.Itisshownthatthereare511typesof NeutroRingsand19171typesofAntiRings.Inparticular,finiteandinfiniteNeutroRingsoftype-NR[8,9]are studied.InNeutroRingsoftype-NR[8,9],theleftandrightdistributiveaxiomsaretakingtobeeitherpartially trueorpartiallyfalseforsomeelements;whileallotherclassicallawsandaxiomsaretakingtobetotally trueforalltheelements.SeveralexamplesandpropertiesofNeutroRingsoftype-NR[8,9]arepresented. NeutroSubrings,NeutroIdeals,NeutroQuotientRingsandNeutroRingHomomorphismsoftheNeutroRingsof type-NR[8,9]arestudiedwithseveralinterestingexamplesandtheirbasicpropertiesarepresented.Itisshown thatinNeutroRingsoftype-NR[8,9],thefundamentaltheoremofhomomorphismsoftheclassicalringsholds. Definition1.1. [8]
(i) Aclassicaloperationisanoperationwelldefinedforalltheset’selements. Doi:10.5281/zenodo.413510087
(ii) ANeutroOperationisanoperationpartiallywelldefined,partiallyindeterminate,andpartiallyouter definedonthegivenset.
(iii) AnAntiOperationisanoperationthatisouterdefinedforallset’selements.
(iv) Aclassicallaw/axiomdefinedonanonemptysetisalaw/axiomthatistotallytrue(i.e.trueforallset’s elements).
(v) ANeutroLaw/NeutroAxiom(orNeutrosophicLaw/NeutrosophicAxiom)definedonanonemptysetisa law/axiomthatistrueforsomeset’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.
(vi) AnAntiLaw/AntiAxiomdefinedonanonemptysetisalaw/axiomthatisfalseforallset’selements.
(vii) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom(axiomthat istrueforsomeelements,indeterminateforotherelements,andfalseforotherelements).
(viii) AnAntiAlgebraisanalgebraendowedwithatleastoneAntiOperationoratleastoneAntiAxiom. Theorem1.2. [7]Let U beanonemptyfiniteorinfiniteuniverseofdiscourseandlet S beafiniteorinfinite subsetof U.If n classicaloperations(lawsandaxioms)aredefinedon S where n ≥ 1,thentherewillbe (2n 1) NeutroAlgebrasand (3n 2n) AntiAlgebras.
2NeutroRingsRevisited
Definition2.1. [Classicalring][6] Let R beanonemptysetandlet +,. : R×R → R bebinaryoperationsoftheusualadditionandmultiplication respectivelydefinedon R.Thetriple (R, +,.) iscalledaclassicalringifthefollowingconditions (R1 R9) hold:
(R1) x + y ∈ R ∀x,y ∈ R [closurelawofaddition].
(R2) x +(y + z)=(x + y)+ z ∀x,y,z ∈ R [axiomofassociativity].
(R3) Thereexists e ∈ R suchthat x + e = e + x = x ∀x ∈ R [axiomofexistenceofneutralelement].
(R4) Thereexists x ∈ R suchthat x +( x)=( x)+ x = e ∀x ∈ G [axiomofexistenceofinverse element]
(R5) x + y = y + x ∀x,y ∈ R [axiomofcommutativity].
(R6) x.y ∈ R ∀x,y ∈ R [closurelawofmultiplication].
(R7) x.(y.z)=(x.y).z ∀x,y,z ∈ R [axiomofassociativity].
(R8) x.(y + z)=(x.y)+(x.z) ∀x,y,z ∈ R [axiomofleftdistributivity].
(R9) (y + z).x =(y.x)+(z.x) ∀x,y,z ∈ R [axiomofrightdistributivity].
Ifinadditionwehave, (R10) x.y = y.x ∀x,y ∈ R [axiomofcommutativity], then (R, +,.) iscalledacommutativering. Definition2.2. [NeutroSophicationofthelawsandaxiomsoftheclassicalring]
(NR1) Thereexistatleastthreeduplets (x,y), (u,v), (p,q) ∈ R suchthat x + y ∈ R (degreeoftruthT)and [u + v = outer-defined/indeterminate(degreeofindeterminacyI)or p + q ∈ R] (degreeoffalsehoodF) [NeutroClosurelawofaddition].
(NR2) Thereexistatleastthreetriplets (x,y,z), (p,q,r), (u,v,w) ∈ R suchthat x +(y + z)=(x + y)+ z (degreeoftruthT)and[[p +(q + r)]or[(p + q)+ r]= outer-defined/indeterminate(degreeofindeterminacyI)or u +(v + w) =(u + v)+ w](degreeoffalsehoodF)[NeutroAxiomofassociativity (NeutroAssociativity)].
Doi:10.5281/zenodo.413510088
(NR3) Thereexistsanelement e ∈ R suchthat x+e = x+e = x and[[x+e]or[e+x]= outer-defined/indeterminate or x + e = x = e + x]foratleastone x ∈ R [NeutroAxiomofexistenceofneutralelement(NeutroNeutralElement)].
(NR4) Thereexists x ∈ R suchthat x+( x)=( x)+x = e and[[ x+x]or[x+( x)]= outer-defined/indeterminate or x + x = e = x +( x)]foratleastone x ∈ R [NeutroAxiomofexistenceofinverseelement(NeutroInverseElement)].
(NR5) Thereexistatleastthreeduplets (x,y), (u,v), (p,q) ∈ R suchthat x + y = y + x and[[p + q]or[q + p]= outer-defined/indeterminate(degreeofindeterminacyI)or u + v = v + u](degreeoffalsehoodF) [NeutroAxiomofcommutativity(NeutroCommutativity)].
(NR6) Thereexistatleastthreeduplets (x,y), (p,q), (u,v) ∈ R suchthat x.y ∈ R (degreeoftruthT)and [u.v = outer-defined/indeterminate(degreeofindeterminacyI)or p.q ∈ R](degreeoffalsehoodF) NeutroClosurelawofmultiplication].
(NR7) Thereexistatleastthreetriplets (x,y,z), (p,q,r), (u,v,w) ∈ R suchthat x.(y.z)=(x.y).z (degreeoftruthT)and[[p.(q.r)]or[(p.q).r]= outer-defined/indeterminate(degreeofindeterminacyI)or u.(v.w) =(u.v).w](degreeoffalsehoodF)[NeutroAxiomofassociativity(NeutroAssociativity)].
(NR8) Thereexistatleastthreetriplets (x,y,z), (p,q,r), (u,v,w) ∈ R suchthat x.(y + z)=(x.y)+(x.z) (degreeoftruthT)and[[p.(q + r)]or[(p.q)+(p.r)]= outer-defined/indeterminate(degreeofindeterminacyI)or u.(v + w) =(u.v)+(u.w)](degreeoffalsehoodF)[NeutroAxiomofleftdistributivity (NeutroLeftDistributivity)].
(NR9) Thereexistatleastthreetriplets (x,y,z), (p,q,r), (u,v,w) ∈ R suchthat (y + z).x =(y.x)+(z.x) (degreeoftruthT)and[[(v + w).u]or[(v.u)+(w.u)]= outer-defined/indeterminate(degreeofindeterminacyI)or (v + w).u =(v.u)+(w.u)](degreeoffalsehoodF)[NeutroAxiomofrightdistributivity (NeutroRightDistributivity)].
(NR10) Thereexistatleastthreeduplets (x,y), (p,q), (u,v) ∈ R suchthat x.y = y.x (degreeoftruthT) and[[p.q]or[q.p]= outer-defined/indeterminate(degreeofindeterminacyI)or u.v = v.u](degreeof falsehoodF)[NeutroAxiomofcommutativity(NeutroCommutativity)].
Definition2.3. [AntiSophicationofthelawandaxiomsoftheclassicalring]
(AR1) Foralltheduplets (x,y) ∈ R, x + y ∈ R [AntiClosurelawofaddition].
(AR2) Forallthetriplets (x,y,z) ∈ R, x +(y + z) =(x + y)+ z [AntiAxiomofassociativity(AntiAssociativity)].
(AR3) Theredoestnotexistanelement e ∈ R suchthat x + e = x + e = x ∀x ∈ R [AntiAxiomofexistence ofneutralelement(AntiNeutralElement)].
(AR4) Theredoesnotexist x ∈ R suchthat x +( x)=( x)+ x = e ∀x ∈ R [AntiAxiomofexistenceof inverseelement(AntiInverseElement)].
(AR5) Foralltheduplets (x,y) ∈ R, x + y = y + x [AntiAxiomofcommutativity(AntiCommutativity)].
(AR6) Foralltheduplets (x,y) ∈ R, x.y ∈ R [AntiClosurelawofmultiplication].
(AR7) Forallthetriplets (x,y,z) ∈ R, x.(y.z) =(x.y).z [AntiAxiomofassociativity(AntiAssociativity)].
(AR8) Forallthetriplets (x,y,z) ∈ R, x.(y + z) =(x.y)+(x.z) [AntiAxiomofleftdistributivity(AntiLeftDistributivity)].
(AR9) Forallthetriplets (x,y,z) ∈ R, (y + z).x =(y.x)+(z.x) [AntiAxiomofrightdistributivity(AntiRightDistributivity)].
(AR10) Foralltheduplets (x,y) ∈ R, x.y = y.x [AntiAxiomofcommutativity(AntiCommutativity)].
Definition2.4. [NeutroRing]
ANeutroRing NR isanalternativetotheclassicalring R thathasatleastoneNeutroLaworatleastoneof {NR1,NR2,NR3,NR4,NR5,NR6,NR7,NR8,NR9} withnoAntiLaworAntiAxiom.
Doi:10.5281/zenodo.413510089
Definition2.5. [AntiRing]
AnAntiRing AR isanalternativetotheclassicalring R thathasatleastoneAntiLaworatleastoneof {AR1,AR2,AR3,AR4,AR5,AR6,AR7,AR8,AR9}
Definition2.6. [NeutroCommutativeRing]
ANeutroNoncommutativeRing NR isanalternativetotheclassicalnoncommutativering R thathasatleast oneNeutroLaworatleastoneof {NR1,NR2,NR3,NR4,NR5,NR6,NR7,NR8,NR9} and NR10 with noAntiLaworAntiAxiom.
Definition2.7. [AntiCommutativeRing]
AnAntiCommutativeRing AR isanalternativetotheclassicalcommutativering R thathasatleastoneAntiLaworatleastoneof {AR1,AR2,AR3,AR4,AR5,AR6,AR7,AR8,AR9} and AR10.
Proposition2.8. Let (R, +,.) beafiniteorinfiniteclassicalring.Then:
(i) Thereare511typesofNeutroRings.
(ii) Thereare19171typesofAntiRings.
Proof. FollowsfromTheorem1.2.
Proposition2.9. Let (R, +,.) beafiniteorinfiniteclassicalcommutativering.Then:
(i) Thereare1023typesofNeutroCommutativeRings.
(ii) Thereare58025typesofAntiCommutativeRings.
Proof. FollowsfromTheorem1.2.
Remark2.10. ItisevidentfromProposition2.8andProposition2.9thattherearemanytypesofNeutroRings andNeutroCommutativeRings.ThetypeofNeutroRingsstudiedbyAgboolain[5]arethoseforwhich NR1 NR10 arealltrue.
Example2.11. (i) Let NR = Z andlet ⊕ beabinaryoperationofordinaryadditionandforall x,y ∈ NR, let beabinaryoperationdefinedon NR as x y = √xy.Then (NR, ⊕, ) isaNeutroRing.
(ii) Let NR = Q andlet ⊕ beabinaryoperationofordinaryadditionandforall x,y ∈ NR,let bea binaryoperationdefinedon NR as x y = x/y.Then (NR, ⊕, ) isaNeutroRing.
(iii) Let AR = N andlet and ⊗ betwobinaryoperationsofordinarysubtractionandordinarymultiplicationrespectivelydefinedon AR.Then (AR, , ⊗) isanAntiRing.
(iv) Let AR = N andlet ⊕ and ⊗ betwobinaryoperationsofordinaryadditionandordinarymultiplication respectivelydefinedon AR.Then (AR, ⊕, ⊗) isanAntiRing.
Definition2.12. Let (NR, +,.) beaNeutroRing.
(i) NR iscalledafiniteNeutroRingoforder n ifthecardinalityof NR is n thatis o(NR)= n.Otherwise, NR iscalledaninfiniteNeutroRingandwewrite o(NR)= ∞
(ii) NR iscalledaNeutroRingwithunityifthereexistsamultiplicativeunitelement u ∈ NR suchthat ux = xu = x foratleastone x ∈ R
(iii) Ifthereexistsaleastpositiveinteger n suchthat nx = e foratleastone x ∈ NR,then NR iscalled aNeutroRingofcharacteristic n.Ifnosuch n exists,then NR iscalledaNeutroRingofcharacteristic zero.
(iv) Anelement x ∈ NR iscalledanidempotentelementif x2 = x.
(v) Anelement x ∈ NR iscalledanilpotentelementiffortheleastpositiveinteger n,wehave xn = e
(vi) Anelement e = x ∈ NR iscalledazerodivisorelementifthereexistsanelement e = y ∈ NR such that xy = e or yx = e
(vii) Anelement x ∈ NR iscalledamultiplicativeinverseelementifthereexistsatleastone y ∈ NR such that xy = yx = u where u isthemultiplicativeunityelementin NR
Doi:10.5281/zenodo.413510090
Definition2.13. Let (NR, +,.) beaNeutroCommutativeRingwithunity.Then
(i) NR iscalledaNeutroIntegralDomainif NR hasnoatleastonezerodivisorelement.
(ii) NR iscalledaNeutroFieldif NR hasatleastonemultiplicativeinverseelement.
Example2.14. Let NR = Z5 = {0, 1, 2, 3, 4} andlet ⊕ and betwobinaryoperationsdefinedon NR by x ⊕ y = x + y 1,x y = x + xy ∀ x,y ∈ NR. Itisclearthat (NR, ⊕) isanabeliangroup.
(1) NeutroAssociativity: Let x,y,z ∈ NR.Then x (y z)= x + xy + xyz, (x y) z = x + xy + xz + xyz, ∴ x + xy + xyz = x + xy + xz + xyz ⇒ xz =0 ∴ x =0 or z =0.
Thisshowsthatonlythetriplets (0,y,z), (x,y, 0), (0,y, 0) canverifyassociativitywith 60% degreeof associativity.
(2) NeutroLeftDistributivity: Let x,y,z ∈ NR.Then x (y ⊕ z)= x + xy + xz x, (x y) ⊕ (x z)= x + xy + x + xz 1, ∴ x + xy + xz x = x + xy + x + xz 1 ⇒ 2x =1 ∴ x =3
Thisshowsthatonlythetriplet (3,y,z) canverifyleftdistributivitywith 20% degreeofleftdistributivity.
(3) NeutroRightDistributivity: Let x,y,z ∈ NR.Then (y ⊕ z) x = y + z 1+ yx + zx x, (y x) ⊕ (z x)= y + yx + z + zx 1, ∴ y + z 1+ yx + zx x = y + yx + z + zx 1 ⇒− x =0 ∴ x =0
Thisshowsthatonlythetriplet (0,y,z) canverifyrightdistributivitywith 20% degreeofrightdistributivity.
(4) NeutroCommutativity: Let x,y ∈ NR.Then x y = x + xy, y x = y + yx, ∴ x + xy = y + yx ⇒ x = y ∴ x = y.
Thisshowsthatonlytheduplet (x,x) canverifycommutativitywith 20% degreeofcommutativity.
WehavejustshownaccordingtoDefinition2.6that (NR, ⊕, ) isaNeutroRing. Example2.15. Let NR = {a,b,c,d} andlet + and bebinaryoperationsdefinedon NR asshownin theCayleytablesbelow: + a b c d a a b c d b b c d a c c d a b d d a b c
a b c d a a b c d b a c b c c c d c d d d a d a
Itisclearthat (NR, +) isanabeliangroup.Fromthetableswehave: Doi:10.5281/zenodo.413510091
(1) NeutroAssociativity:
a(bc)=(ab)c = b, b(bb)= b but (bb)b = d = b.
ThisshowsNeutroAssociativityof +
(2) NeutroLeftDistributivity:
a(b + c)= ab + ac = d, b(c + d)= c but bc + bd = d = c.
ThisshowsNeutroLeftDistributivityof over +
(3) NeutroRightDistributivity:
(b + c)c = bc + cc = d, (b + c)a = d but ba + ca = c = d.
ThisshowsNeutroRightDistributivityof over +
(4) NeutroCommutativity: ac = ca = a, bc = b but cb = d = b.
ThisshowsNeutroCommutativityof
WehavejustshownaccordingtoDefinition2.6that (NR, +,.) isaNeutroRing.
Example2.16. FromExample2.15,wenotethat e = a istheadditiveneutralelement.Wenowhavethe following:
(i) NR isaNeutroCommutativeRingwithunitysince aa = a,ac = ca = c,ad = da = d.
(ii) {a,c} areidempotentelements.
(iii) {d} isanilpotentelement.
(iv) {b,d} arezerodivisorelements.
(v) {a,d} areinvertibleelements.
(vi) NR isnotaNeutroIntegralDomain.
(vii) NR isaNeutroField.
(viii) NR isaNeutroCommutativeRingofcharacteristic2.
Example2.17. Let U = {e,a,b,c,d,f } beauniverseofdiscourseandlet NR = {e,a,b,c}.Supposethat ∗ and ◦ aretwobinaryoperationsdefinedon NR asshownintheCayleytablesbelow:
◦ e a b c e e a b c a a b or e c b b b c c or e a c c b a f
∗ e a b c e e b c a or b or e a a c e d b b e a c c c a b e
Itisclearthat (NR, ◦) isaNeutroGroup.Nowconsiderthefollowing: Doi:10.5281/zenodo.413510092
(i) NeutroAssociativityof ∗: a ∗ (b ∗ b)=(a ∗ b) ∗ b = c, b ∗ (a ∗ b)= b but (b ∗ a) ∗ b = c = b, a ∗ (b ∗ c)= d (outer-defined), (a ∗ b) ∗ c = e ∗ c = indeterminate.
(ii) NeutroLeftDistributivityof ∗ over ◦: e ∗ (e ◦ e)=(e ∗ e) ◦ (e ∗ e)= e, a ∗ (b ◦ e)= e but (a ∗ b) ◦ (a ∗ e)= a = e, a ∗ (b ◦ c)= e but (a ∗ b) ◦ (a ∗ c)= e ◦ d =?
(iii) NeutroRightDistributivityof ∗ over ◦: (e ◦ e) ∗ e)=(e ∗ e) ◦ (e ∗ e)= e, (b ◦ c) ∗ a = c but (b ∗ a) ◦ (c ∗ a)= a = c, (e ◦ e) ∗ c = e ∗ e =? and (e ∗ c) ◦ (e ∗ c)=?
(iv) NeutroCummutativityof ∗: e ∗ e = c ∗ c = e, b ∗ c = c but c ∗ b = b = c, e ∗ c = indeterminatebut c ∗ e = c. Hence (NR, ◦, ∗) isaNeutroRing.
3FiniteandInfiniteNeutroRingsofType-NR[8,9]
Inthissection,wearegoingtostudyatypeofNeutroRings (NR, ◦, ∗) where R1,R2,R3,R4,R5,R6,R7,R10 aretotallytrueforalltheelementsof NR,andwhere R8 and R9 areeitherpartiallytrueorpartiallyfalsefor someelementsof NR.ThistypeofNeutroRingswillbecalledNeutroRingsoftype-NR[8,9].
Example3.1. Let NR = Z6 = {0, 1, 2, 3, 4, 5} andlet ◦ and ∗ betwobinaryoperationsdefinedon NR by x ◦ y = x + y,x ∗ y = x + y + xy ∀x,y ∈ NR where + isadditionmodulo6.Then (NR, ◦, ∗) isafiniteNeutroRingoftype-NR[8,9].Toseethis,consider theCayleytablesbelow. ◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4
∗ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 3 5 1 3 5 2 2 5 2 5 2 5 3 3 1 5 3 1 5 4 4 3 2 1 0 5 5 5 5 5 5 5 5
Itisclearfromthetablesthat (NR, ◦) isanabeliangroupwith e =0 astheidentityelement,andthat (NR, ∗) isacommutativesemigroup.ItremainstoshowthatthetwodistributiveaxiomsareNeutroAxioms.
(i) NeutroLeftDistributivityof ∗ over ◦: Let x,y,z ∈ NR.Then x ∗ (y ◦ z)= x + y + z + xy + xz and (x ∗ y) ◦ (x ∗ z)=2x + y + z + xy + xz.Forleftdistributivitytohold,wemusthave 2x = x from whichweobtain x =0.Hence,onlythetriplets (0,y,z), (0,y, 0), (0, 0,z), (0, 0, 0) canverifytheleft distributivityof ∗ over ◦ with66.67%degreeofdistributivity.Hence, ∗ isNeutroLeftDistributiveover ◦ in NR
(ii) NeutroRightDistributivityof ∗ over ◦: Itcansimilarlybeshownthat ∗ isNeutroRightDistributive over ◦ with66.67%degreeofdistributivity. Hence, (NR, ◦, ∗) isafiniteNeutroRingoftype-NR[8,9].
Example3.2. Let NR = Z or Q or R or C andlet ◦ and ∗ betwobinaryoperationsdefinedon NR by x ◦ y = x + y,x ∗ y = x + y + xy ∀x,y ∈ NR where + istheordinaryadditionofintegersorrationalsorrealsorcomplexnumbers.Itisclearthat (NR, ◦) isanabeliangroupwith e =0 astheidentityelement,andthat (NR, ∗) isacommutativesemigroup. ItremainstoshowthatthetwodistributiveaxiomsareNeutroAxioms.
(i) NeutroLeftDistributivityof ∗ over ◦: Let x,y,z ∈ NR.Then x ∗ (y ◦ z)= x + y + z + xy + xz and (x ∗ y) ◦ (x ∗ z)=2x + y + z + xy + xz.Forleftdistributivitytohold,wemusthave 2x = x from whichweobtain x =0.Hence,onlythetriplets (0,y,z), (0,y, 0), (0, 0,z), (0, 0, 0) canverifytheleft distributivityof ∗ over ◦.Hence, ∗ isNeutroLeftDistributiveover ◦ in NR Doi:10.5281/zenodo.413510093
(ii) NeutroRightDistributivityof ∗ over ◦: Itcansimilarlybeshownthatthat ∗ isNeutroRightDistributive over ◦
Hence, (NR, ◦, ∗) isaninfiniteNeutroRingoftype-NR[8,9].
Proposition3.3. Let (NRi, ◦, ∗),i =1, 2 beNeutroRingsoftype-NR[8,9].IntheCartesianproduct NR1 × NR2 of NRi,let ⊕ and betwobinaryoperationsdefined ∀(w,x), (y,z) ∈ NR1 × NR2 asfollows:
(w,x) ⊕ (y,z)=(w ◦ y,x ◦ z) (w,x) (y,z)=(w ∗ y,x ∗ z)
Then (NR1 × NR2, ⊕, ) isaNeutroRingoftype-NR[8,9].
Proof. FollowsfromDefinition2.2.
Proposition3.4. Let (NR, +,.) beaNeutroRingoftype-NR[8,9]andlet e betheidentityelementin NR with respectto + .Thenforsome x,y,z ∈ NR,wehave:
(i) x.e = e.
(ii) x.( y) = (x.y) =( x).y
(iii) x.(y z) = x.y x.z
(iv) (y z).x = y.x z.x.
Proof. Since isNeutroDistributiveover + ,therequiredresultsfollow.
Proposition3.5. Let (NR, +,.) beaNeutroRingoftype-NR[8,9].Thenforsome w,x,y,z ∈ NR,wehave:
(i) (w + x).(y + z) =(w.y + w.z)+(x.y + x.z).
(ii) (w + x) (y z) =(w.y + x.y) (w.z + x.z)
(iii) (w x) (y z) =(w.y + x.z) (w.z + x.y)
(iv) (w + x).(w x) =(w.w x.x)+(x.w w.x).
Proof. Since isNeutroDistributiveover + ,therequiredresultsfollow.
Proposition3.6. Let (NR, +,.) beaNeutroRingoftype-NR[8,9]andlet m,n ∈ N.Then ∀x ∈ NR,we have:
(i) xm.xn = xm+n
(ii) (xm)n =(xn)m = xmn .
Proof. Since isassociative,therequiredresultsfollow.
Definition3.7. Let (NR, ◦, ∗) beaNeutroRingoftype-NR[8,9]andlet NS benonemptysubsetof NR
(i) NS iscalledaNeutroSubringof NR if (NS, ◦, ∗) isalsoaNeutroRingoftype-NR[8,9].
(ii) NS iscalledaQuasiNeutroSubringof NR if (NS, ◦, ∗) isaNeutroRingofthetypedifferentfromthe typeoftheparentNeutroRing NR
TheonlytrivialNeutroSubringof NR is NR
Proposition3.8. ThereexistNeutroRingsoftype-NR[8,9]withonlytrivialNeutroSubrings.
Proof. Considerthestructure (NR, ◦, ∗) suchthat NR = Z6 and ∀ x,y ∈ NR,wehave x ◦ y = x + y +1,x ∗ y = x + y +3xy andconsiderthestructure (NS, ◦, ∗) where NS = Z and ∀ x,y ∈ NS, x ◦ y = x + y 7,x ∗ y = x + y 3xy.Itcanbeshownthat NR and NS areNeutroRingsoftype-NR[8,9] withonlytrivialNeutroSubrings.
Doi:10.5281/zenodo.413510094
Example3.9. Let (NR, ◦, ∗) betheNeutroRingofExample3.1andlet NS1 = {0, 3} and NS2 = {0, 2, 4} be twosubsetsof NR.Itcaneasilybeshownthat (NS1, ◦, ∗) and (NS2, ◦, ∗) areNeutroRingsoftype-NR[8,9] andconsequentlytheyareNeutroSubringsof NR.Itisobservedthat NS1 ∩ NS2 = {0} and NS1 ∪ NS2 = {0, 2, 3, 4} arenotNeutroSubringsof NR.Also, NS1 × NS2 = {(0, 0), (0, 2), (0, 4), (3, 0), (3, 3), (3, 4)} is aNeutroSubringof NR × NR.
Example3.10. Let (NR, ◦, ∗) betheNeutroRingofExample3.2andlet NS1 =2Z, NS2 =3Z and NS3 =4Z bethreesubsetsof NR.Itcaneasilybeshownthat NS1,NS2 and NS3 areNeutroSubringsof NR.Generallyforpositiveintegers n ≥ 2,itcanbeshownthat NS = nZ areNeutroSubringsof NR.It isobservedthat NS1 ∩ NS2 =6Z, NS1 ∩ NS3 =4Z, NS2 ∩ NS3 =12Z and NS1 ∪ NS3 =2Z are NeutroSubringsof NR.However, NS1 ∪ NS2 and NS2 ∪ NS3 arenotNeutroSubringsof NR
Proposition3.11. Let (NR, ◦, ∗) beaNeutroRingoftype-NR[8,9]andlet {NSi},i =1, 2 beNeutroSubrings of NR.Then
(i) NS = NS1 ∩ NS2 isnotnecessarilyaNeutroSubringof NR.
(ii) NS = NS1 × NS2 isaNeutroSubringof NR × NR
(iii) NS = NS1 ∪ NS2 isnotnecessarilyaNeutroSubringof NR
Definition3.12. Let (NR, ◦, ∗) beaNeutroRingoftype-NR[8,9].Anonemptysubset NI of NR iscalleda NeutroIdealof NR ifthefollowingconditionshold:
(i) NI isaNeutroSubringof NR.
(ii) x ∈ NI and r ∈ NR implythatatleastone r ∗ x or x ∗ r ∈ NI forall r ∈ NR
Definition3.13. Let (NR, ◦, ∗ ) beaNeutroRingoftype-NR[8,9].Anonemptysubset NI of NR iscalleda QuasiNeutroIdealof NR ifthefollowingconditionshold:
(i) NI isaQuasiNeutroSubringof NR.
(ii) x ∈ NI and r ∈ NR implythatatleastone x ∗ r or r ∗ x ∈ NI forall r ∈ NR
Example3.14. Let NI1 = NS1 = {0, 3} and NI2 = NS2 = {0, 2, 4} beNeutroSubringsofExample 3.9.Thenfor NI1,wehave 0 ∗ 0=0, 1 ∗ 0=1, 2 ∗ 0=2, 3 ∗ 0=3, 4 ∗ 0=4, 5 ∗ 0=5 and 0 ∗ 3=3, 1 ∗ 3=1, 2 ∗ 3=5, 3 ∗ 3=3, 4 ∗ 3=1, 5 ∗ 3=5.Accordingly, NI1 isaNeutroIdeal. Alsofor NI2,wehave 0 ∗ 0=0, 1 ∗ 0=1, 2 ∗ 0=2, 3 ∗ 0=3, 4 ∗ 0=4, 5 ∗ 0=5, 0 ∗ 2=2, 1 ∗ 2= 5, 2 ∗ 2=2, 3 ∗ 2=5, 4 ∗ 2=2, 5 ∗ 2=5 and 0 ∗ 4=4, 1 ∗ 4=3, 2 ∗ 4=2, 3 ∗ 4=1, 4 ∗ 4=0, 5 ∗ 4=5 Accordingly, NI2 isaNeutroIdeal.
Example3.15. Let NI1 = NS1 =2Z, NI2 = NS2 =3Z and NI3 = NS3 =4Z beNeutroSubringsof Example3.10.Itcaneasilybeshownthat NI1, NI2 and NI3 areNeutroIdeals.Generally, NI = nZ are NeutroIdealsfor n ≥ 2 Definition3.16. Let (NR, ◦, ∗) beaNeutroRingoftype-NR[8,9]andlet NI beaNeutroIdealof NR.The set NR/NI isdefinedby
NR/NI = {x ◦ NI : x ∈ NR}
For x ◦ NI,y ◦ NI ∈ NR/NI with x,y ∈ NR,let ⊕ and bebinaryoperationson NR/NI definedas follows: (x ◦ NI) ⊕ (y ◦ NI)=(x ◦ y) ◦ NI, (x ◦ NI) (y ◦ NI)=(x ∗ y) ◦ NI.
Ifthetriple (NR/NI, ⊕, ) isaNeutroRingoftype-NR[8,9],itwillbecalledaNeutroQuotientRing.
Example3.17. Let NI1 = {0, 3} and NI2 = {0, 2, 4} beNeutroIdealsofExample3.14.For NI1,wehave NR/NI1 = {NI1, 1+ NI1, 2+ NI1}
Doi:10.5281/zenodo.413510095
andthecompositionsofelementsof NR/NI1 accordingtoDefinition3.16aregivenintheCayleytables:
⊕ NI1 1+ NI1 2+ NI1 NI1 NI1 1+ NI1 2+ NI1 1+ NI1 1+ NI1 2+ NI1 NI1 2+ NI1 2+ NI1 NI1 1+ NI1
NI1 1+ NI1 2+ NI1 NI1 NI1 1+ NI1 2+ NI1 1+ NI1 1+ NI1 NI1 2+ NI1 2+ NI1 2+ NI1 2+ NI1 2+ NI1
ItcaneasilybededucedfromtheCayleytablesthat (NR/NI1, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI1 astheidentityelement. For NI2,wehave
NR/NI2 = {NI2, 1+ NI2}
andthecompositionsofelementsof NR/NI2 accordingtoDefinition3.16aregivenintheCayleytables:
⊕ NI2 1+ NI2 NI2 NI2 1+ NI2 1+ NI2 1+ NI2 NI2
NI2 1+ NI2 NI2 NI2 1+ NI2 1+ NI2 1+ NI2 1+ NI2
ItcaneasilybededucedfromtheCayleytablesthat (NR/NI2, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI2 astheidentityelement. Example3.18. Let NI1 =2Z, NI2 =3Z and NI3 =4Z beNeutroIdealsofExample3.15.For NI1,we have NR/NI1 = {NI1, 1+ NI1} andthecompositionsofelementsof NR/NI1 accordingtoDefinition3.16aregivenintheCayleytables:
⊕ NI1 1+ NI1 NI1 NI1 1+ NI1 1+ NI1 1+ NI1 NI1
NI1 1+ NI1 NI1 NI1 1+ NI1 1+ NI1 1+ NI1 1+ NI1
ItcaneasilybededucedfromtheCayleytablesthat (NR/NI1, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI1 astheidentityelement.
For NI2,wehave NR/NI2 = {NI2, 1+ NI2, 2+ NI2} andthecompositionsofelementsof NR/NI2 accordingtoDefinition3.16aregivenintheCayleytables:
⊕ NI2 1+ NI2 2+ NI2 NI2 NI2 1+ NI2 2+ NI2 1+ NI2 1+ NI2 2+ NI2 NI2 2+ NI2 2+ NI2 NI2 1+ NI2
NI2 1+ NI2 2+ NI2 NI2 NI2 1+ NI2 2+ NI2 1+ NI2 1+ NI2 NI2 2+ NI2 2+ NI2 2+ NI2 2+ NI2 2+ NI2
ItcaneasilybededucedfromtheCayleytablesthat (NR/NI2, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI2 astheidentityelement. For NI3,wehave NR/NI3 = {NI3, 1+ NI3, 2+ NI3, 3+ NI3} andthecompositionsofelementsof NR/NI3 accordingtoDefinition3.16aregivenintheCayleytables:
⊕ NI3 1+ NI3 2+ NI3 3+ NI3 NI3 NI3 1+ NI3 2+ NI3 3+ NI3 1+ NI3 1+ NI3 3+ NI3 1+ NI3 3+ NI3 2+ NI3 2+ NI3 1+ NI3 NI3 3+ NI3 3+ NI3 3+ NI3 3+ NI3 3+ NI3 3+ NI3
NI3 1+ NI3 2+ NI3 3+ NI3 NI3 NI3 1+ NI3 2+ NI3 3+ NI3 1+ NI3 1+ NI3 2+ NI3 3+ NI3 NI3 2+ NI3 2+ NI3 3+ NI3 NI3 1+ NI3 3+ NI3 3+ NI3 NI3 1+ NI3 2+ NI3
ItcaneasilybededucedfromtheCayleytablesthat (NR/NI3, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI3 astheidentityelement.
Doi:10.5281/zenodo.413510096
Proposition3.19. Let (NR, +,.) beaNeutroRingoftype-NR[8,9]andlet NI beaNeutroIdealof NR.For x+NI,y +NI ∈ NR/NI with x,y ∈ NR,let ⊕ and bebinaryoperationson NR/NI definedasfollows: (x + NI) ⊕ (y + NI)=(x + y)+ NI, (x + NI) (y + NI)=(xy)+ NI.
Thenthetriple (NR/NI, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI astheidentityelement.
Proof. Supposethat (NR, +,.) isaNeutroRingoftype-NR[8,9]andsupposethat NI isNeutroIdealof NR. Thatthebinaryoperations ⊕ and on NR/NI arewell-definedarethesameasfortheclassicalrings.Itis clearthat (NR/NI, ⊕) isanabeliangroupwith e = NI astheidentityelementandthat (NR/NI, ) isa commutativesemigroup.Since NR isoftype-NR[8,9],itfollowsthatthereexistsatleastatriplet (x,y,z) ∈ NR suchthat x(y + z) = xy + xz and (y + z)x = yx + zx.Consequently, (x + NI) ((y + NI) ⊕ (z + NI))= x(y + z)+ NI =(xy + xz)+ NI =[(x + NI) (y + NI)] ⊕ [(x + NI) (z + NI)] and ((y + NI) ⊕ (z + NI)) (x + NI)=(y + z)x + NI =(yx + zx)+ NI =[(y + NI) (x + NI)] ⊕ [(z + NI) (x + NI)]
Hence, (NR/NI, ⊕, ) isaNeutroRingoftype-NR[8,9]with e = NI astheidentityelement. Definition3.20. Let (NR, +,.) and (NS, + ,. ) beanytwoNeutroRingsoftype-NR[8,9].Themapping φ : NR → NS iscalledaNeutroRingHomomorphismif φ preservesthebinaryoperationsof NR and NS thatisifforatleastaduplet (x,y) ∈ NR,wehave:
φ(x + y)= φ(x)+ φ(y), φ(x.y)= φ(x) φ(y)
Thekernelof φ denotedby Kerφ isdefinedas
Kerφ = {x : φ(x)= eNR}
Theimageof φ denotedby Imφ isdefinedas Imφ = {y ∈ NS : y = φ(x) foratleastone y ∈ NS}.
Ifinaddition φ isaNeutroBijection,then φ iscalledaNeutroRingIsomorphismandwewrite NR ∼ = NS NeutroRingEpimorphism,NeutroRingMonomorphism,NeutroRingEndomorphismandNeutroRingAutomorphismaredefinedsimilarly.
Example3.21. Let (NR, +, ∗) betheNeutroRingofExample3.1.
(i) Let φ : NR → NR beamappingdefinedby
φ(x)=2 ∗ x ∀x ∈ NR.
Then, φ isnotaNeutroRingHomomorphism.Since ∗ isNeutroDistributiveover + ,wehavefor x,y ∈ NR,
φ(x + y)=2 ∗ (x + y) =2 ∗ x +2 ∗ y = φ(x)+ φ(y)
Thisshowsthat φ doesnotpreserve + .Howeversince ∗ isassociativeand 2 ∗ 2=2,wehave ∀x,y ∈ NR
φ(x ∗ y)=2 ∗ (x ∗ y) =(2 ∗ x) ∗ (2 ∗ y) = φ(x) ∗ φ(y)
Thisshowsthat φ preserves ∗.Accordingly, φ isnotaNeutroRingHomomorphism.
Doi:10.5281/zenodo.413510097
(ii) Let φ : NR × NR → NR beaprojectiondefinedby
φ(x,y)= x ∀x,y ∈ NR.
Itcaneasilybeshownthat φ isaNeutroRingHomomorphismwith Kerφ = {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5)} and Imφ = {0, 1, 2, 3, 4, 5} = NR.
Itcanbeshownthat Kerφ isaNeutroIdealof NR × NR.
Example3.22. Let NR/NI1 = {NI1, 1+ NI1, 2+ NI1} betheNeutroQuotientRingofExample3.17and let φ : NR → NR/NI1 beamappingdefinedby φ(x)= x + NI1 ∀x ∈ NR.Then
φ(0)= φ(3)= NI1, φ(1)= φ(4)=1+ NI1, φ(2)= φ(5)=2+ NI1
Itcaneasilybeshownthat φ isaNeutroRingHomomorphismwith Kerφ = {0, 3} = NI1
Example3.23. Let NR/NI3 = {NI3, 1+ NI3, 2+ NI3, 3+ NI3} betheNeutroQuotientRingofExample 3.18andlet φ : NR → NR/NI3 beamappingdefinedby φ(x)= x + NI3 ∀x ∈ NR.Then
φ(0)= NI3, φ(1)=1+ NI3, φ(2)=2+ NI3, φ(3)=3+ NI3
Itcaneasilybeshownthat φ isaNeutroRingHomomorphismwith Kerφ =4Z = NI3
Proposition3.24. Let NR and NS betwoNeutroRingsoftype-NR[8,9]andsupposethat φ : NR → NS is aNeutroRingHomomorphism.Then:
(i) φ(eNR)= eNS .
(ii) Kerφ isaNeutroIdealof NR
(iii) Imφ isaNeutroSubringof NS
(iv) φ isNeutroInjectiveifandonlyif Kerφ = {eNR}.
Proof. Theproofisthesameasfortheclassicalringsandsoomitted.
Proposition3.25. Let NI beaNeutroIdealoftheNeutroRing NR oftype-NR[8,9].Themapping ψ : NR → NR/NI definedby ψ(x)= x + NI ∀x ∈ NR
isaNeutroRingEpimomorphismandthe Kerψ = NI .
Proof. Theproofisthesameasfortheclassicalringsandsoomitted.
Proposition3.26. [FundamentalTheoremofNeutroRingHomomorphisms]. Let NR and NS beNeutroRingsoftype-NR[8,9]andlet φ : NR → NS beaNeutroRingHomomorpismwith K = Kerφ.Thenthe mapping ψ : NR/K → Imφ definedby
ψ(x + K)= φ(x) ∀x ∈ NR
isaNeutroRingIsomorphism.
Proof. Theproofisthesameasfortheclassicalringsandsoomitted.
Doi:10.5281/zenodo.413510098
4Conclusion
WehaveinthispaperrevisitedtheconceptofNeutroRingsintroducedbyAgboolain[5].Itwasshownthat thereare511typesofNeutroRingsand19171typesofAntiRings.Inparticular,wehavestudiedfiniteandinfiniteNeutroRingsoftype-NR[8,9].IntheclassofNeutroRingsoftype-NR[8,9],theleftandrightdistributive axiomsweretakingtobeeitherpartiallytrueorpartiallyfalseforsomeelements;whileallotherclassicallaws andaxiomsweretakingtobetotallytrueforalltheelements.SeveralexamplesandpropertiesofNeutroRings oftype-NR[8,9]werepresented.NeutroSubrings,NeutroIdeals,NeutroQuotientRingsandNeutroRingHomomorphismsoftheNeutroRingsoftype-NR[8,9]werestudiedwithseveralinterestingexamplesandtheirbasic propertieswerepresented.ItwasshownthatintheclassofNeutroRingsoftype-NR[8,9],thefundamental theoremofhomomorphismsoftheclassicalringsholds.
5Acknowledgment
TheauthorisverygratefultoProfessorFlorentinSmarandacheforhisvaluableprivatediscussionsonthe paper.Theusefulcommentsandsuggestionsoftheanonymousreviewersareacknowledged.
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