Let be a commutative ring with identity, and let be any ideal of . Define the radical RADICAL of I be the set for some positive integer .
Prove that is an ideal of .
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 I think this 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?