Let be a commutative ring with identity, and let be any ideal of . Define the radicalRADICALof I be the set for some positive integer .

Prove that is an ideal of .

My Attempt:

I think I should use the "ideal test"; that is to show that (i) whenever , and (ii) show that and are in whenever and .

(i) Let , then that means such that . Since is an ideal of , , and Ithinkthis implies that .

But first I think we must show that . How do we show this?

(ii) Since I is an ideal of R we know that for any and , and are both in . Is this correct?