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1、w.b.vasanthakandasamysmarandachenear-ringsamericanresearchpressrehoboth2002SmarandacheNear-ringsW.B.VasanthaKandasamyDepartmentofMathematicsIndianInstituteofTechnology,MadrasChennai–600036,IndiaAmericanResearchPressRehoboth,NM20021Thepictureonthecoverisarepresentatio
2、noftheSmarandachenear-ring(S-near-ring).Theinternationalnear-ringcommunityrepresentsthenear-ringbyastructurewhichisnearlyaring,butnotexactlyaring,i.e.anear-ring.FortheS-near-ring,ofcourse,wehavea"S"withinthenear-ringstructure!Thisbookcanbeorderedinapaperboundreprintf
3、rom:BooksonDemandProQuestInformation&Learning(UniversityofMicrofilmInternational)300N.ZeebRoadP.O.Box1346,AnnArborMI48106-1346,USATel.:1-800-521-0600(CustomerService)http://wwwlib.umi.com/bod/andonlinefrom:PublishingOnline,Co.(Seattle,WashingtonState)at:http://Publis
4、hingOnline.comThisbookhasbeenpeerreviewedandrecommendedforpublicationby:Prof.GeoffreyL.Booth,UniversityofPortElizabeth,P.O.1600,SouthAfrica;Prof.V.Tharmarathnam,Dept.ofMathematics,Univ.ofJaffna,SriLanka.Prof.L.TuÛescu,Dept.ofMathematics,FraÛiiBuzeÕtiCollege,Romania.C
5、opyright2002byAmericanResearchPressandW.B.VasanthaKandasamyRehoboth,Box141NM87322,USAManybookscanbedownloadedfrom:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htmISBN:1-931233-66-7StandardAddressNumber:297-5092PrintedintheUnitedStatesofAmerica2CONTENTSP
6、reface5ChapterOne:PREREQUISITES1.1Groupswithexamples71.2Definitionofgroupoidsandloopswithexamples81.3Semigroups131.4Semirings141.5Latticesanditsproperties15ChapterTwo:NEAR-RINGSANDITSPROPERTIES2.1Definitionofnear-ringandsomeofitsbasicproperties192.2N-groups,homomorph
7、ismandideal-likesubsets212.3Products,directsumsandsubdirectproductinnear-rings222.4Idealsinnear-rings232.5Modularityinnear-rings252.6Nearpolynomialrings262.7Nearmatrixrings27ChapterThree:SPECIALCLASSESOFNEAR-RINGSANDTHEIRGENERALIZATIONS3.1IFPnear-rings293.2Groupnear-
8、ringsanditsgeneralizations353.3Loopnear-ringsanditsgeneralizations413.4Groupoidnear-ringsanditsproperties543.5Specialpropertiesofne
