a characterization of the radical of an ideal

Proposition 1.

Let I be an ideal in a ring R, and I be its radicalPlanetmathPlanetmathPlanetmath. Then I is the intersectionMathworldPlanetmath of all prime idealsMathworldPlanetmathPlanetmath containing I.


Suppose xI, and P is a prime ideal containing I. Then R-P is an m-system (http://planetmath.org/MSystem). If xR-P, then (R-P)I, contradicting the assumptionPlanetmathPlanetmath that IP. Therefore xR-P. In other words, xP, and we have one of the inclusions.

Conversely, suppose xI. Then there is an m-system S containing x such that SI=. Enlarge I to a prime ideal P disjoint from S, so that xP (we can do this; for a proof, see the second remark in this entry (http://planetmath.org/MSystem)). By contrapositivity, we have the other inclusion. ∎

Remark. This shows that every prime ideal is a radical ideal: for P is the intersection of all prime ideals containing P, and if P is itself prime, then P=P.

Title a characterization of the radical of an ideal
Canonical name ACharacterizationOfTheRadicalOfAnIdeal
Date of creation 2013-03-22 18:04:50
Last modified on 2013-03-22 18:04:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Derivation
Classification msc 16N40
Classification msc 13-00
Classification msc 14A05