Let alpha : R ---> S be a ring homomorphism

(i) If R is a principal ideal domain (PID) and S is an integral domain. Show that alpha is EITHER injective OR Im(alpha) is a field.

(ii) Let S be a integral domain. By considering a ring homomorphism e_0: R = S[x]----> S by f(x) ---> f(0). Show that if R=S[x] is a PID, then S is a field.

Ok, in part (i), we have to prove that Im(alpha) is field if and only if Ker(alpha) is a maximal ideal of R. But I don't know how to prove it.

Part (ii) looks a bit hard to me, I dont know where to start

Any ideas would be greatful

Thank you so much