Let be a commutative ring and let be an ideal. Define the radical of , , by . Prove that is an ideal of containing and that .
First, we certainly have since for all , . In particular, is nonempty. Now let with . Now by this previous exercise. Note that if , then , and if , then , since is an ideal. Thus every term in the expansion of is in , and thus . We also have, for all , , and in particular . Thus is an ideal of .
Now note the following. if and only if , if and only if for some , if and only if in for some , if and only if in for some , if and only if . Thus .