Let be a commutative ring with . Prove that if is prime, then , where denotes the nilradical of . Deduce that is contained in the intersection of all prime ideals of .
Let be a prime ideal, and let be nilpotent with . Let be minimal such that . Note that in , which is an integral domain. If , we have , so that is a zero divisor in , a contradiction. Thus we have , and . Thus , and moreover if denotes the collection of all prime ideals of , we have .