Examples and non-examples The principal ideals in the (commutative) ring are In fact, every ideal of is principal (see ). In any ring , the sets and are principal ideals. For any ring and element the ideals and are respectively left, right, and two-sided principal ideals, by definition. For example, is a principal ideal of In the commutative ring of complex polynomials in two variables, the set of polynomials that vanish everywhere on the set of points is a principal ideal because it can be written as (the set of polynomials divisible by ). In the same ring , the ideal generated by both and is not principal. (The ideal is the set of all polynomials with zero for the constant term.) To see this, suppose there was a generator for so Then contains both and so must divide both and Then must be a nonzero constant polynomial. This is a contradiction since but the only constant polynomial in is the zero polynomial. In the ring the numbers where is even form a non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider and These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are and they are not associates.
exAmPLEs anD nON-eXampLeS The prinCipaL idEals in THe (ComMutATive) ring are in facT, EveRy ideal of is princIpal (See ). in AnY riNG , tHe setS ANd aRe prinCiPal Ideals. for anY ring and eleMEnt the ideALS and are rEspecTively Left, Right, and Two-SideD priNcipal IdeAls, by deFinITiON. fOr exAmple, is a principAl IdEAl Of In the comMuTaTIve riNg Of CompleX polynomIaLs IN twO varIables, thE Set of pOlYnoMials THat VAnish everyWHerE on thE SET oF pointS iS a prIncipal iDEal becAuSe it can be wrItTEN AS (the Set Of polynOmiAls DIViSibLe By ). In thE sAMe RIng , The iDeal GeneRated By bOth and is Not principal. (tHe IdEal Is the set Of all poLynoMiaLs witH Zero For tHe CONStanT TeRm.) tO see ThIS, suppose therE wAS a GeNErator fOR SO then ContaiNS both and so mUSt Divide both aNd Then musT be A nonZeRO CoNsTaNt polYnomiAL. this is a coNTradictIon since but tHE only coNStant polynOmiaL in Is tHe zEro POlYnomiaL. in thE RiNg THe NuMbeRs WHERe is eVEn fOrm a non-pRinCipal IDeal. tHis iDeal formS A ReGulaR HEXAgonal laTtICE IN the comPlex PlAne. CONSIder and these numBErs aRE eLementS Of this iDeAL with The same Norm (Two), BuT BECAuse The ONlY uNItS in the rIng ARe and TheY aRE nOt asSociates.