ideals contained in a union of radical ideals

Let R be a commutative ring and IR an ideal. Recall that the radicalPlanetmathPlanetmathPlanetmathPlanetmath of I is defined as


It can be shown, that r(I) is again an ideal and Ir(I). Let

V(I)={PR|P is a prime ideal and IP}.

Of course V(I) (because I is contained in at least one maximal idealMathworldPlanetmathPlanetmath) and it can be shown, that


Finaly, recall that an ideal I is called radical, if I=r(I).

PropositionPlanetmathPlanetmathPlanetmath. Let I,R1,,Rn be ideals in R, such that each Ri is radical. If


then there exists i{1,,n} such that IRi.

Proof. Assume that this not true, i.e. for every i we have IRi. Then for any i{1,,n} there exists PiV(Ri) such that IPi (this follows from the fact, that Ri=r(Ri) and the characterizationMathworldPlanetmath of radicals via prime idealsMathworldPlanetmathPlanetmath). But for any i we have RiPi and thus


ContradictionMathworldPlanetmathPlanetmath, since each Pi is prime (see the parent object for details).

Title ideals contained in a union of radical ideals
Canonical name IdealsContainedInAUnionOfRadicalIdeals
Date of creation 2013-03-22 19:04:23
Last modified on 2013-03-22 19:04:23
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Corollary
Classification msc 13A15