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1.Basicalgebraicpropertiesofintegers
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studiesofdifferenttheoriesofnumbersandcomparingthemisaninterest-
ingdirectionofresearch,animportantpartofthefoundationsofarithmetic.
2.BasicstructuresinthesetNofnaturalnumbers
AllwaysofdefiningnaturalnumbersleadtoasetNofnaturalnumbers
withtwobinaryoperations(additionandmultiplication)andanorder≤.
Thebinaryoperationsdefineastructureofacommutativesemiringwith1
and0andwithanorder≤.TheInductionAxiomcanbeformulated(and
isoftenapplied)as:
♦EverynonemptysubsetX⊂Ncontainsanelementsmallestinthatset.
InthesemiringNwehaveanalgorithmofdividingwitharemainderand
analgorithmoffindingthegreatestcommondivisor(gcd).Onecaneasily
defineinNthenotionofaprimenumberandprovethebasictheoremof
arithmeticofthenaturalnumberssayingthat:
♦Everynaturalnumbern>1canbeexpressedasaproductofprimenum-
bersandsuchadecompositionisuniqueuptotheorderofthefactors.
Theproof(byinduction)ofthefirstpartiseasy;theproofofthesecond
partissomewhatcumbersomeandismucheasierafterextendingNtothe
ringZofintegers.
3.TheringZ
TheformaldefinitionoftheringZandoftheembeddingN→Zcanbe
describedinthefollowingway.
InN×Nwedefineanequivalencerelation≡by
(m1,n1)≡(m2,n2)iffm1+n2=m2+n1
anddefineZtobethesetofequivalenceclassesof≡.Denotetheequivalence
classcontaining(m,n)by[(m,n)].NextdefineanembeddingN→Zby
nl→[(n,0)]anddefinemultiplicationandadditioninZby
[(m1,n1)]+[(m2,n2)]=[(m1+m2,n1+n2)],
[(m1,n1)]·[(m2,n2)]=[(m1m2+n1n2,m2n1+m1n2)].
ThenZbecomesacommutativeringwith1containingNasasubsemiring
(weidentifyeverynaturalnumberwithitsimageinZ).TheringZisa
domain(doesnotcontainzerodivisors).
