Let I be an ideal of a commutative ring R with identity. Prove that the radical of I is an ideal of R and each prime ideal that contains I, also contains . Also prove that is the nilradical of

Proof.

Now the radical of I is

So pick and , so , since I is an ideal. Therefore is also an ideal.

Now, suppose that Q is a prime ideal of R such that . Pick , so then such that So I need to show that ... How should I start?