__RADICAL__of I to be the set N(I) = {[tex]r \in R[/tex]: [tex]r^n \in I[/tex] for some positive integer n}.

I need the proof that:

- If I contains a unit of R, then show that I = R.

- N(N(I))=N(I)

An integral domain is a commutative, unital ring that contains no zero-divisors. So I'm guesing R must be an ID. If R is a ring with identity 1

_{R}, then [tex]a \in R[/tex] is a unit if ab = ba = 1

_{R}for some b in R, and b is the inverse of a. Now if we suppose I contains a, how do we show that I=R?

For the second proof we can assume N(I) is an ideal of R.

Any help or suggestions are appreciated.