Direct proof that, in a commutative ring with only one prime ideal $P$, every element of $P$ is nilpotent 
Let $R$ be a commutative ring with identity such that $R$ has exactly one prime ideal $P$.
Prove: all elements in $P$ are nilpotent.

While doing this problem, I used the fact that "the nilradical of $R$ is equal to the intersection of all prime ideals of $R$" (in this case, the intersection of all prime ideals $=P$), then I can solve this problem.
However, it seems to me that this fact overkills this problem because we have a strong condition that there is only one prime ideal.
I am here to ask if there is a much more simple and direct approach (which I have overlooked) to solving this problem.

Let me put it in another way: is there any proof without using Zorn's lemma?
 A: When reading this post I began to wonder if $Nil(R)=\bigcap\{ P\mid P\text{ prime}\}$ was equivalent to choice, but that led me to this interesting post.
If I read it correctly, the nilradical equation is not equivalent to AC!
Hope this helps!
A: with this assumption one can show more:  all  element  of  $R$  is  invertible  or  nilpotent .
A: I would try something like the following. Let $a$ be in $P$ and suppose that $a$ is not nilpotent. Let $S$ be the multiplicative set of powers of $a$. Then $S$ does not contain $0$. By Zorn (I don't know if this is an acceptable strategy for you), there is a prime ideal $Q$, maximal with respect to not intersecting $S$. But then $Q$ cannot equal $P$, contradicting the fact that $P$ is the unique prime of $R$. Thus, $a$ is nilpotent.
This argument is basically a specialization of the result you quoted (see, e.g., p. 148 of Lang, Algebra, 1971 edition). Mohamed's result that if $a$ is not nilpotent then it must be a unit is pretty straightforward. Incidently, Theorem 1 of Kaplansky's "Commutative Rings" is the statement that an ideal $I$ that is maximal with respect to the property that it is disjoint from a multiiplicatively closed set $S$ is prime. Of course, one needs a way to get such a maximal element, e.g., Zorn. I don't know if Zorn can be avoided completely in this situation.  
