Let be a commutative ring. Recall that an element is called nilpotent if for some nonzero natural number . Prove that the set of all nilpotent elements in form an ideal. is called the nilradical of . [Hint: Use the Binomial Theorem to prove closure under addition.]
Let . Then for some nonnegative natural numbers and , we have .
Consider . By the Binomial Theorem, we have . Note that if , , and if , then . Thus , and we have . Moreover, we have , so that . Since , is an additive subgroup of .
Finally, since is commutative, if then and likewise . Thus absorbs on the left and the right, and hence is an ideal.