# Math Help - Nilradical in prime ideals

1. ## Nilradical in prime ideals

Let R be a commutative ring, prove that each prime ideal of R contains the nilradical of R.

Proof so far.

Let I be a prime ideal of R, then if $xy \in I$, we would have either x or y in I.

Now, nilradical of R is an element $r \in R$ such that $r^n=0 \ \ \ n \geq 1$.

How should approach this problem? I don't see how prime would lead to zero here. Thanks.

2. Is $0$ in $I$?

3. Yes.

So $0^1 = 0$.... Is that right? That sounds almost too easy...

4. What you want to prove is that $x \in Nil(R) \Rightarrow x \in I$.
But if $x \in Nil(R)$, there is a $n \in \mathbb{N}$ such that $x^{n}=0$