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

a=m1+r1xS;b=m2+r2yS;

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

ab=(m1+r1x)(m2+r2y)=m1m2+m2r1x+m1r2y+xyr1r2.

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