ideals contained in a union of radical ideals
Let R be a commutative ring and I⊆R an ideal. Recall that the radical of I is defined as
r(I)={x∈R|∃n∈ℕxn∈I}. |
It can be shown, that r(I) is again an ideal and I⊆r(I). Let
V(I)={P⊆R|P is a prime ideal and I⊆P}. |
Of course V(I)≠∅ (because I is contained in at least one maximal ideal) and it can be shown, that
r(I)=⋂P∈V(I)P. |
Finaly, recall that an ideal I is called radical, if I=r(I).
Proposition. Let I,R1,…,Rn be ideals in R, such that each Ri is radical. If
I⊆R1∪⋯∪Rn, |
then there exists i∈{1,…,n} such that I⊆Ri.
Proof. Assume that this not true, i.e. for every i we have I⊈. Then for any there exists such that (this follows from the fact, that and the characterization of radicals via prime ideals
). But for any we have and thus
Contradiction, since each 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 |