multiplicative sets in rings and prime ideals

PropositionPlanetmathPlanetmath. Let R be a commutative ring, SR a mutliplicative subset of R such that 0S. Then there exists prime idealMathworldPlanetmathPlanetmathPlanetmath PR such that PS=.

Proof. Consider the family 𝒜={IR|I is an ideal and IS=}. Of course 𝒜, because the zero idealMathworldPlanetmathPlanetmath 0𝒜. We will show, that 𝒜 is inductive (i.e. satisfies Zorn’s Lemma’s assumptionsPlanetmathPlanetmath) with respect to inclusion.

Let {Ik}kK be a chain in 𝒜 (i.e. for any a,bK either IaIb or IbIa). Consider I=kKIk. Obviously I is an ideal. Furthermore, if xIS, then there is kK such that xIkS=. Thus IS=, so I𝒜. Lastely each IkI, which completesPlanetmathPlanetmathPlanetmathPlanetmath this part of proof.

By Zorn’s Lemma there is a maximal elementMathworldPlanetmath P𝒜. We will show that this ideal is prime. Let x,yR be such that xyP. Assume that neither xP nor yP. Then PP+(x) and PP+(y) and these inclusions are proper. Therefore both P+(x) and P+(y) do not belong to 𝒜 (because P is maximal). This implies that there exist a(P+(x))S and b(P+(y))S. Thus


where m1,m2P and r1,r2R. Note that abS. We calculate


Of course m1m2,m2r1x,m1r2yP, because m1,m2P and xyr1r2P by our assumption that xyP. This shows, that abP. But abS and P𝒜. ContradictionMathworldPlanetmathPlanetmath.

Corollary. Let R be a commutative ring, I an ideal in R and SR a multiplicative subset such that IS=. Then there exists prime ideal P in R such that IP and PS=.

Proof. Let π:RR/I be the projection. Then π(S)R/I is a multiplicative subset in R/I such that 0+Iπ(S) (because IS=). Thus, by proposition, there exists a prime ideal P in R/I such that Pπ(S)=. Of course the preimageMathworldPlanetmath of a prime ideal is again a prime ideal. Furthermore Iπ-1(P). Finaly π-1(P)S=, because Pπ(S)=. This completes the proof.

Title multiplicative sets in rings and prime ideals
Canonical name MultiplicativeSetsInRingsAndPrimeIdeals
Date of creation 2013-03-22 19:03:58
Last modified on 2013-03-22 19:03:58
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Theorem
Classification msc 16U20
Classification msc 13B30