Suppose we have an integral domain, S, of characteristic p, where p is prime. If we have the ring homomorphism θ: S → S by θ(a) = a^p, prove that θ is onto.
Since the commutative ring S is arbitrary, my plan is to show that θ is one-to-one, and since it maps S to S, it must also be onto. However, I'm not sure how to show this. Any help would be greatly appreciated.
Edit: Disregarding whether θ is onto, how could you show that Kerθ contains only 0, so that you could make the statement that θ is one-to-one?