For part (a), let be an ideal in . What can you say about ?
(a) Suppose that is a surjective ring homomorphism. Prove that if is a principal ideal domain then every ideal in is also principal.
(b) Give an example to show that S need not be an integral domain.
Well yes, but this isn't the best way to argue. You want to show that or equivalently that if then for some .
Right.Conclusion: J is principal..
OK, but what's an obvious example of a ring which is a PID, one that's not, and an obvious homorphism between the two? Hint: is involved.Then has at least one nonzero zero-divisor. So if then if , and can both be nonzero.