every prime ideal is radical

Let β„› be a commutative ring and let 𝔓 be a prime idealMathworldPlanetmathPlanetmath of β„›.

Proposition 1.

Every prime ideal P of R is a radical ideal, i.e.


Recall that π”“βŠŠβ„› is a prime ideal if and only if for any a,bβˆˆβ„›

aβ‹…bβˆˆπ”“β‡’aβˆˆπ”“β’Β or ⁒bβˆˆπ”“

Also, recall that

Rad⁑(𝔓)={rβˆˆβ„›βˆ£βˆƒnβˆˆβ„•β’Β such that ⁒rnβˆˆπ”“}

Obviously, we have π”“βŠ†Rad⁑(𝔓) (just take n=1), so it remains to show the reverse inclusion.

Suppose r∈Rad⁑(𝔓), so there exists some nβˆˆβ„• such that rnβˆˆπ”“. We want to prove that r must be an element of the prime ideal 𝔓. For this, we use inductionMathworldPlanetmath on n to prove the following propositionPlanetmathPlanetmathPlanetmath:

For all nβˆˆβ„•, for all rβˆˆβ„›, rnβˆˆπ”“β‡’rβˆˆπ”“.

Case n=1: This is clear, rβˆˆπ”“β‡’rβˆˆπ”“.

Case n β‡’ Case n+1: Suppose we have proved the proposition for the case n, so our induction hypothesis is


and suppose rn+1βˆˆπ”“. Then


and since 𝔓 is a prime ideal we have

rβˆˆπ”“β’Β or ⁒rnβˆˆπ”“

Thus we conclude, either directly or using the induction hypothesis, that rβˆˆπ”“ as desired.


Title every prime ideal is radical
Canonical name EveryPrimeIdealIsRadical
Date of creation 2013-03-22 13:56:54
Last modified on 2013-03-22 13:56:54
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 13-00
Classification msc 14A05
Synonym prime ideal is radical
Related topic RadicalOfAnIdeal
Related topic PrimeIdeal
Related topic HilbertsNullstellensatz