AppliedMathematicalSciences,Vol.8,2014,no.164,8227-8232HIKARILtd,www.m - PDF document

AppliedMathematicalSciences,Vol.8,2014,no.164,8227-8232HIKARILtd,www.m-hikari.comhttp://dx.doi.org/10.12988/ams.2014.410840PrimenessofExtensionofSemi-idealsinPosetsJ.CatherineGraceJohnandB.Elavarasan1DepartmentofMathematics,KarunyaUniversityCoimbatore-641114,Tamilnadu,IndiaCopyrightc 2014J.CatherineGraceJohnandB.Elavarasan.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunre-stricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.AbstractInthisshortnote,weobtainequivalentconditionsforextensionofsemi-diealsofPtobeminimalprimesemi-idealsofP:MathematicsSubjectClassication:06D60Keywords:Poset,semi-ideal,primeideal,minimalprimeideal1PreliminariesThroughoutthispaper(P,)denotesaposetwithsmallestelement0:Forbasicterminologyandnotationforposets,werefer[5]and[4].ForMP;letL(M)=fx2P:xmforallm2MgdenotesthelowerconeofMinPanddually,letU(M)=fx2P:mxforallm2MgbetheupperconeofMinP:LetA;BP,weshallwriteL(A;B)insteadofL(A[B)andduallyfortheuppercones.IfM=fx1;x2;:::;xngisnite,thenweusethenotationL(x1;x2;:::;xn)insteadofL(fx1;x2;:::;xng)(anddually).ItisclearthatforanysubsetAofP,wehaveAL(U(A))andAU(L(A)):IfAB,thenL(B)L(A)andU(B)U(A):Moreover,LUL(A)=L(A)andULU(A)=U(A):Following[6],anonemptysubsetIofPiscalledsemi-idealifb2Iandab;thena2I:Apropersemi-idealIofPiscalled 1Correspondingauthor 8228J.CatherineGraceJohnandB.ElavarasanprimeifL(a;b)Iimpliesthateithera2Iorb2I[4].Foranysemi-idealIofPandasubsetAofP,wedeneA;I&#x-388;=fz2P:L(a;z)Iforalla2Ag:ItisclearthatA;I&#x-417;=\a2Aa;I&#x-417;:IfA=fxg;thenwewritefxg;I&#x]TJ/;༔ ;.9;Ւ ;&#xTf 1;.42; 0 ;&#xTd [;=x;I&#x-278;:Foranysemi-idealIofP;itiseasytoverifythatA;I&#x-278;;I;&#x-278;;I&#x-278;=A;I&#x-278;foranysubsetAofP:Following[5],asubsetIofPiscalledidealifforanya;b2I;wehaveL(U(a;b))I.ItisclearthateveryidealofPissemi-ideal,butconverseneednotbetrueingeneral.ItisalsoclearthattheintersectionoftwoidealsofPisagainanidealofP;butthefollowingexampleshowsthatunionoftwoidealsofPisnotnecessarytobeanidealofP:Example1.1ConsiderP=f0;a;b;c;d;eganddenearelationonPasfollows. Then(P;)isaposetandA=f0;agandB=f0;bgareidealsofP:HereA[B=f0;a;bgisnotanidealofPasL(U(a;b))*A[B:2MainResultsTheorem2.1LetIbeasemi-idealofPandB,Cbenon-emptysubsetsofP:Thenthefollowingstatementshold.(i)IfBC,thenC;I&#x-278;B;I&#x-278;.(ii)B[C;I&#x-278;=B;I&#x-278;\C;I&#x-278;.(iii)B;I&#x-278;\C;I&#x-278;B\C;I&#x-278;.(iv)IfJisasemi-idealofPandIJ,thenB;I&#x-278;B;J&#x-278;.(v)IfI1;I2;:::;Inaresemi-idealsofP;thenB;n\i=1Ii&#x-278;=n\i=1B;Ii&#x-278;.(vi)B;I&#x-278;\BI.(vii)IfI1,I2aresemi-idealsofPwithI1\I2I;thenI1I2;I&#x-280;andI2I1;I&#x-278;.(viii)IfBC;I&#x-278;orCB;I&#x-278;;thenB\CI:(ix)B;I&#x-278;=B;B;I&#x-278;&#x-278;. Primenessofextensionofsemi-idealsinposets8229Proof:(i)Letx2C;I&#x-278;.ThenL(x;c)Iforallc2C;SinceBC;wehavex2B;I&#x-278;.(ii)Letx2B[C;I&#x-278;.ThenL(x;y)Iforally2B[Cwhichimpliesx2B;I&#x-354;\C;I&#x-354;.Forconverse,letx2B;I&#x-354;\C;I&#x-354;:ThenL(x;y)Iforally2B[Cwhichimpliesx2B[C;I&#x-278;.(iii)Letx2B;I&#x-278;\C;I&#x-278;.ThenL(x;b)IandL(x;c)Iforallb2Bandc2C.Inparticular,wehaveL(x;t)Iforallt2B\C.(iv)Letx2B;I&#x-278;.ThenL(x;y)IJforally2B.(v)Itisclearthatx2B;n\i=1Ii&#x-382;ifandonlyifL(x;y)n\i=1Iiforally2Bifandonlyifx2n\i=1B;Ii&#x-278;.(vi)Letx2B;I&#x-341;\B.ThenL(x;y)Iforally2B:Sincex2B;wehavex2I.(vii)Letx2I1andsupposex=2I2;I&#x-410;.ThenL(x;c)*Iforsomec2I2.ButL(x;c)I1\I2I;acontradiction.(viii)SupposeBC;I&#x-323;andletx2B\C.ThenL(x;c)Iforallc2Cwhichimpliesx2I.(ix)Letx2B;B;I&#x-296;&#x-296;.ThenL(x;b)B;I&#x-297;forallb2B.Nowforanyb2B;letr2L(x;b).ThenL(r;b)I.Sincerb,wehaver2I;soL(x;b)Iforanyb2B.Thusx2B;I&#x-278;.Forasemi-idealIofPandasubsetBofP;B;I&#x-311;ismaximalamongthesetfA;I&#x-297;:APandA;I&#x-297;6=PgifandonlyifB;I&#x-297;6=PandB;I&#x-380;C;I&#x-380;6=PimpliesB;I&#x-381;=C;I&#x-380;foranysubsetCofP[2].ItisclearthatthemaximalelementsamongthesetfA;I&#x-376;:APandA;I&#x-395;6=Pgisoftheformb;I&#x-395;forsomeb2P=I.Asemi-idealminimalinthesetofallprimesemi-idealscontainingforsomegivensemi-idealIiscalledminimalprimesemi-idealofPcontainingI.Theorem2.2LetIbeasemi-idealofPandBasubsetofP.Thenthefollowingconditionsareequivalent.(i)B;I&#x-364;isamaximalelementamongthesetfA;I&#x-364;:APandA;I&#x-278;6=Pg.(ii)B;I&#x-278;isaprimesemi-idealofP.(iii)B;I&#x-278;isaminimalprimesemi-idealofPcontainingI.Proof:(i))(ii)LetB;I&#x-507;beamaximalelementamongthesetfA;I&#x]TJ ;&#x-387;&#x.745;&#x -14;&#x.446;&#x Td ;&#x[000;:AA;I&#x-288;6=Pg.Thenthereexistsb2B=IsuchthatB;I&#x-288;=b;I&#x]TJ ;&#x-387;&#x.745;&#x -14;&#x.446;&#x Td ;&#x[000;.LetL(x;y)b;I&#x-286;andsupposethatx=2b;I&#x-286;forsomex;y2I.Theny2x;b;I&#x-503;&#x-503;andthereexistst2L(x;b)=Isuchthatx;b;I&#x]TJ ;&#x-360;&#x.729;&#x -14;&#x.446;&#x Td ;&#x[000;&#x]TJ ;&#x-360;&#x.729;&#x -14;&#x.446;&#x Td ;&#x[000;t;b;I&#x-387;&#x-387;:Wenowclaimthatt;I&#x-387;=b;I&#x-387;.Lets2x;b;I&#x-438;&#x-438;andr2L(s;t).ThenL(r;b)I:Sincertb;wehave 8230J.CatherineGraceJohnandB.Elavarasanr2IwhichimpliesL(s;t)I;thuss2t;I&#x-463;andhenceb;I&#x-463;x;b;I&#x-328;&#x-328;t;I&#x-328;.Sincet;I&#x-328;6=Pandb;I&#x-328;ismaximal,wehavet;I&#x-278;=b;I&#x-278;.Soy2x;b;I&#x-278;&#x-278;t;I&#x-278;=b;I&#x-278;.(ii))(iii)LetB;I&#x-303;andQbeprimesemi-idealofPwithIQB;I&#x]TJ ;&#x-387;&#x.746;&#x -14;&#x.445;&#x Td ;&#x[000;.SinceB;I&#x-422;6=P:;thereexistsy2B=IsuchthatB;I&#x-422;y;I&#x]TJ ;&#x-387;&#x.746;&#x -14;&#x.446;&#x Td ;&#x[000;6=P:Lett2B;I&#x-390;.ThenL(t;y)IQ.SinceQisprimeandy=2Q;wehavet2Q.(iii))(ii)Itistrivial.(ii))(i)LetB;I&#x-532;beaprimesemi-idealofPandB;I&#x-532;C;I&#x]TJ ;&#x-387;&#x.745;&#x -14;&#x.446;&#x Td ;&#x[000;6=PforsomesubsetCofP.Thenthereexistsy2CnIsuchthatC;I&#x-530;=y;I&#x-530;.Leta2C;I&#x-530;:ThenL(a;y)IB;I&#x-530;.SinceB;I&#x-406;isaprimesemi-idealandy=2I;wehavea2B;I&#x-406;.SoC;I&#x-278;=B;I&#x-278;.Theorem2.3IfIisapropersemi-idealofP;thenPhasdistinctminimalprimesemi-idealsQj=yj;I&#x-278;ofP;whereyj2PnI;with\Qj=I.Proof:LetS=fA;I&#x-528;:APandA;I&#x-528;6=Pgandy2PnI:ThenS6=fgasy;I&#x-388;2S.ByZorn'slemmaShasmaximalelements.LetQj=yj;I&#x-375;(j2J)bethedistinctmaximalelementsamongthesetfA;I&#x-278;:APandA;I&#x-278;6=Pg;whereyj2PnI.ThenbyTheorem2.2,Qj=yj;I&#x-278;areminimalprimesemi-idealsofP.Wenowclaimthat\j2JQj=I.ClearlyI\j2JQj.Letx=2I:Thenx;I&#x-325;6=Pandx;I&#x-325;Qjforsomej.Ifx2\j2JQj;thenx2yj;I&#x-325;whichimpliesyj2x;I&#x-278;Qj=yj;I&#x-278;;acontradiction.Thusx=2\j2JQj.Therefore\j2JQj=I.LetIbeasemi-idealofP:AposetPsatisesconditionifwheneverL(A;B)IimpliesAB;I&#x-278;foranysubsetsAandBofP.InExample1.1,letA=f0;a;b;cg;B=f0;bgandI=f0;a;dg:ThenL(A;B)I;butA*B;I&#x-427;=f0;a;dg.Sothereexistsaposetswhichnotsatisescondition.Following[1],anidealIofPiscalledstronglyprimeifL(A;B)IimpliesthateitherAIorBIforanyidealsA;BofP;whereA=A�f0g.AnidealIofaposetPiscalledstronglysemi-primeifL(A;B)IandL(A;C)ItogetherimplyL(A;U(B;C))IforallidealsA;BandCofP.ItisclearbyExample2.5of[1]thatA;I&#x-301;isnotanidealofPforsubsetAofP;butwehavethefollowing.Theorem2.4LetIbeasemiprimeidealofPandBasubsetofP.IfPsatisesthecondition;theneverymaximalelementamongthesetfA;I&#x-278;:APandA;I&#x-278;6=PgisstronglyprimeidealofP. Primenessofextensionofsemi-idealsinposets8231Proof:LetB;I&#x-278;beamaximalelementamongthesetfA;I&#x-278;:APandA;I&#x-278;6=Pg.ThenbyTheorem2.2andProposition15of[5],B;I&#x-278;isprimeidealofP.LetAandCbeidealsofPwithL(A;C)B;I&#x-371;andsupposethatA*B;I&#x-278;.Thenthereexistsa2AnB;I&#x-278;suchthatL(a;c)B;I&#x-278;forallc2Cwhichimpliesc2B;I&#x-278;.Corollary2.5([4],Theorem6)LetIbeasemi-idealofP.ThenI=\fJ:J2Spec(I)g;whereSpec(I)isthesetofallprimesemi-idealsofPcontainingI.Corollary2.6LetIbeasemiprimeidealofPandBasubsetofP.ThenPhasdistinctminimalprimeidealsQi=yi;I&#x-278;ofP;whereyi2PnI;with\Qi=I.Proof:ItfollowsfromProposition15of[5]andTheorem2.3.Lemma2.7LetIbeapropersemi-idealofPandAP:Thentheascendingchainconditionanddescendingchainconditiononsemi-idealsoftheformA;I&#x-278;coincide.Proof:SupposeascendingchainconditionholdsforsemiidealoftheformA;I&#x-409;;foranyAP.LetA1;I&#x-409;A2;I&#x-409;A3;I&#x-409;:::beadescendingchainofsemiidealsofPforsubsetAjofPforj=1;2;3;::::ThenA1;I&#x-313;;I&#x-313;A2;I&#x-313;;I&#x-313;A3;I&#x-313;;I&#x-313;:::isaassentingchainofsemi-idealsofP;whichterminatesafteranitenumberofstepsbythegivencondition.ThisinturnimpliesthatthedescendingchainA1;I&#x-343;;I&#x-343;;I&#x-343;A2;I&#x-451;;I&#x-451;;I&#x-451;A3;I&#x-451;;I&#x-451;;I&#x-451;:::ortheoriginalchainterminatesafteranitenumberofsteps.Theconverseissimilar.LetIbeanidealofP.ThenPsatisesmaximum-IconditionifandonlyifforeverysubsetBofP;B;I&#x-410;hasamaximalelementamongthesetfA;I&#x-278;:APandA;I&#x-278;6=Pg.Theorem2.8LetIbeapropersemi-idealofP.Thenthefollowingcondi-tionsareequivalent:(i)Psatisesmaximum-Icondition.(ii)Phasonlynitenumberofdistinctminimalprimesemi-idealsQj=yj;I&#x]TJ ;&#x-387;&#x.746;&#x -23;&#x.91 ;&#xTd [;;whereyj2PnI)ofI;forj=1;2;3;:::;n;n\j=1Qj=I.Proof:(i))(ii)ByTheorem,Phasdistinctminimalprimesemi-idealsQj=yj;I&#x-400;ofP;whereyj2PnI;with\j2JQj=I.WenowclaimthatjJjisnite.Ifnot,thenforsomej12J;yj1;I&#x-400;isnotcontainedinallyj;I&#x-278;forj2J. 8232J.CatherineGraceJohnandB.ElavarasanTakesomej22J;yj1;I&#x-490;*yj2;I&#x-490;whichimpliesyj1;I&#x-490;yj1;I&#x]TJ ;&#x-387;&#x.746;&#x -14;&#x.446;&#x Td ;&#x[000;\yj2;I&#x-410;.Ifyj1;I&#x-410;\yj2;I&#x-410;6=I;thenthisprocesscanbecontinuedandsoon,wecangetadescendingchainyj1;I&#x-306;yj1;I&#x-306;\yj2;I&#x-401;::::Sinceyj1;I&#x-401;\yj2;I&#x-401;=I;fj1;j2g&#x-401;;wehaveadescendingchainyj1;I&#x-450;yj1;I&#x-450;\yj2;I&#x-450;::::notterminated,whichiscontradictiontoLemma2.7.SojJjisnite.(i))(ii)ItfollowsfromTheorem2.2.Corollary2.9LetIbeapropersemiprimeidealofP.IfPsatisesthecondition;thenthefollowingconditionsareequivalent:(i)Psatisesmaximum-Icondition.(ii)PhasonlynitenumberofdistinctminimalstronglyprimeidealsQj=yj;I&#x-278;;whereyj2PnIofP;forj=1;2;3;:::;n;n\j=1Qj=I.Proof:ItfollowsfromProposition15of[5]andTheorem2.8.References[1]J.CatherineGraceJohnandB.Elavarasan,StronglyPrimeandStronglySemiprimeidealsinPosets,submitted.[2]W.H.CornishandP.N.Stewart,Ringswithnonilpotentelementsandwiththemaximumconditiononannihilators,Canad.Math.Bull.,17(1),35-38(1974).http://dx.doi.org/10.4153/cmb-1974-006-1[3]A.K.GoyalandS.C.Choudhary,Near-ringswithnonon-zeronilpo-tenttwo-sidedR-subsets,PeriodicaMathematicaHungarica,20(2),161-167(1989).http://dx.doi.org/10.1007/bf01848154[4]RadomirHalas,Onextensionofidealsinposets,DiscreateMathematics,308,4972-4977(2008).http://dx.doi.org/10.1016/j.disc.2007.09.022[5]V.S.KharatandK.A.Mokbel,Primenessandsemiprimenessinposets,Math.Bohem.,134(1),19-30(2009).[6]P.V.Venkatanarasimhan,Semiidealsinposets,Math.Ann.,185(4),338-348(1970).http://dx.doi.org/10.1007/bf01349957Received:October21,2014;Published:November24,2014