Show that nilpotent elements of a commutative ring R constitute an ideal of R.

Proof.

So nilpotent element is an element $\displaystyle a \in R $ such that $\displaystyle a^n =0 $ for some n>0.

Denote the set of all nilpotent elements of R by N.

Pick $\displaystyle a \in N $ and $\displaystyle r \in R $

We need to show that $\displaystyle ar \in N $

$\displaystyle (ar)^n=a^nr^n=0$

That's it? Thanks.