Hello once again
In the first statement, you have probably seen statements like:
is a prime ideal of if and only if is an integral domain.
This is because the condition [TEX]ab\in P[TEX] implies or
becomes implies or in , as we are modding out by elements of P.
Similarly the primary condition becomes the condition that all zero divisors are nilpotent and it is a similar proof to the statement above. By passing to the quotient ring R/P, and using the ideal correspondence theorem, we see P corresponds to the zero ideal in R/P. So it suffices to prove the statement for the zero ideal in R/P.
Now the radical of the zero ideal, actually consists of all nilpotent elements, by definition.
For the second point, we appeal to the statement that requires Zorn's Lemma: Any proper ideal is contained in some maximal ideal. So P must be contained in some maximal ideal Q. All maximal ideals are prime, so Q is prime and Q contains P. As P is the only prime it follows that P=Q and so P is maximal.