So what exactly are prime ideals? They are ideals such that or .
Let , where is a commutative ring. How is the Zariski topology motivated?
Let be the set of all prime ideals which contain . All ideals that contain contain the whole of . Hence all members of also contain the whole of .
Now take , where is any arbitrary two-element set in . Clearly . Note that for both these properties, there is nothing special that is true only for prime ideals and not for any other ideal containing the elements.
Now take , where is the ideal generated by the element . Here is the radical of . It can be proved that . The inclusion of in the equality relation is what sets apart prime ideals containing from any random ideal containing .
Now we will go one step further. Let be the prime ideals containing . We will prove that . It is known that the nilradical of is the intersection of all prime ideals in it. Now consider prime ideals in , where . The intersection of all prime ideals in will again be the nilradical of . The inverse of all such prime ideals will be prime ideals containing in . The rest of the argument is easy to see from here.