This question has been put on hold although it is very precisely formulated and evokes an interesting and underestimated property of rings.
The question is intimately related to a useful concept in scheme theory: the maximal points (introduced in Grothendieck and Dieudonné's EGA), which generalize the non always present generic point of a scheme.
There is no proof that the question is homework, although this is taken for granted in the explanation for the on-holding it.
The OP's imperative tone ("show that...") is a little rough, but we are mathematicians, not thin-skinned aristocrats at the court of Louis XIV, and we should not let this seriously bother us .
Finally, that someone has no activity in or knowledge of a topic does not mean that a related question should be called off-topic.
I have voted to reopen the question and would like to reemphasize how idiotic and nefarious is the (out of our control, I know) rule that five persons with absolutely no proven competence in a subject can on-hold a question on a site with about 65000 registered users.
Sadly the answer and comments to my question are in the same vein as the reasons of the onholders : they consist in general formal statements on how a question should be formulated and on the fact that I'm ranting. Yes, I am ranting because the question should not have been put on hold.
There is zero consideration for the mathematics involved, which is strange on a mathematical site which is not officially a site on étiquette.
To repeat: rings with just one prime ideal are interesting and useful in algebraic geometry. The OP asks why they are the same as rings whose nilradical is a prime. This is a good question, exactly at the level of this site. My ranting question: is there a mathematical argument for putting it on hold ?
The question has just been reopened. Thanks to all the reopeners (with one obvious exception).
Even newer but hopefully last edit
I have had three downvotes on main today , which is the record in the 34 months I heve been registered on math.stackexchange (but the day is not over yet...).
One of them was for this answer which had had 21 upvotes and no other downvote since it was posted on the 21st of March 2012.
I don't think that this downvote, nor the others, are independent of this discussion.
I don't really care so much about reputation (although being a human being, I confess it has a certain vanity appeal) but that kind of behaviour considerably cools my desire for any further discussion here and I will now go back to my dear geometry on main, enjoy the great questions and answers there and try to contribute if I can.