ideals contained in a union of radical ideals
It can be shown, that is again an ideal and . Let
Finaly, recall that an ideal is called radical, if .
Proposition. Let be ideals in , such that each is radical. If
then there exists such that .
Proof. Assume that this not true, i.e. for every we have . 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|
|Date of creation||2013-03-22 19:04:23|
|Last modified on||2013-03-22 19:04:23|
|Last modified by||joking (16130)|