Prove that if is a commutative ring and is a finitely generated ideal, where and each is nilpotent, then is a nilpotent ideal. Deduce that if the nilradical of is finitely generated then it is a nilpotent ideal.

Say for each . Now let .

By this previous exercise, we know that , where . Then it must be the case that for each element of , some is at least . Thus , and we have . Thus is a nilpotent ideal.

If is finitely generated, then each generator is (by definition) nilpotent. Thus is a nilpotent ideal.