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1、w.b.vasanthakandasamysmarandacheringsa18aa213aa108a16a5a6a17a11a7a9a1aa14a4a15a123a0americanresearchpress2002SmarandacheRingsW.B.VasanthaKandasamyDepartmentofMathematicsIndianInstituteofTechnology,MadrasChennai–600036,IndiaAmericanResearchPressRehoboth,NM20021Thepictu
2、reonthecoveristhelatticerepresentationoftheS-idealsoftheSmarandachemixeddirectproductringR=Z3×Z12×Z7.ThisisamajordifferencebetweenaringandaSmarandachering.For,inaringthelatticerepresentationofidealsisalwaysamodularlatticebutweseeincaseofS-ringsthelatticerepresentation
3、ofS-idealneednotingeneralbemodular.Thisbookcanbeorderedinapaperboundreprintfrom:BooksonDemandProQuestInformation&Learning(UniversityofMicrofilmInternational)300N.ZeebRoadP.O.Box1346,AnnArborMI48106-1346,USATel.:1-800-521-0600(CustomerService)http://wwwlib.umi.com/bod/
4、andonlinefrom:PublishingOnline,Co.(Seattle,WashingtonState)at:http://PublishingOnline.comPeerreviewers:IonGoian,DepartmentofAlgebraandNumberTheory,UniversityofKishinev,Moldova.SebastianMartinRuiz,Avda.DeRegla,43,Chipiona11550(Cadiz),Spain.DwirajTalukdar,Head,Departmen
5、tofMathematics,NalbariCollege,Nalbari,Assam,India.Copyright2002byAmericanResearchPressandW.B.VasanthaKandasamyRehoboth,Box141NM87322,USAManybookscanbedownloadedfrom:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htmISBN:1-931233-64-0StandardAddressNumber:2
6、97-5092PrintedintheUnitedStatesofAmerica2CONTENTSPreface51.Preliminarynotions1.1Groups71.2Semigroups111.3Lattices141.4Smarandachesemigroups192.Ringsanditsproperties2.1Definitionandexamples212.2Specialelementsinrings242.3Substructuresofaring262.4Homomorphismandquotient
7、rings282.5Specialrings302.6Modules342.7Ringswithchainconditions353.Smarandacheringsanditsproperties3.1DefinitionofSmarandacheringwithexamples383.2Smarandacheunitsinrings413.3Smarandachezerodivisorsinrings463.4Smarandacheidempotentsinrings513.5SubstructuresinS-rings563
8、.6Smarandachemodules633.7RingssatisfyingS.A.C.C.andS.D.C.C.673.8Somespecialtypesofrings683.9SpecialelementsinS-rings713.10Sp
