every prime ideal is radical
Every prime ideal of is a radical ideal, i.e.
Recall that is a prime ideal if and only if for any
Also, recall that
Obviously, we have (just take ), so it remains to show the reverse inclusion.
For all , for all , .
Case : This is clear, .
Case Case : Suppose we have proved the proposition for the case , so our induction hypothesis is
and suppose . Then
and since is a prime ideal we have
Thus we conclude, either directly or using the induction hypothesis, that as desired.
|Title||every prime ideal is radical|
|Date of creation||2013-03-22 13:56:54|
|Last modified on||2013-03-22 13:56:54|
|Last modified by||alozano (2414)|
|Synonym||prime ideal is radical|