1. ## Ring homomorphism

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

2. 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.
You may have in your course something (first isomorphism theorem?) like: if $\displaystyle f$ is a ring morphism between two rings $\displaystyle A$ and $\displaystyle B,$ then $\displaystyle f(A)\cong A/\text{ker}f$
The fact that for any ring $\displaystyle R,$ the quotient of $\displaystyle R$ by one of its ideal $\displaystyle I$ is a field iff $\displaystyle I$ is a maximal ideal will allow you to conclude.

ii) So a way to prove that $\displaystyle S$ is a field is to show that $\displaystyle e_0$ is surjective homomorphism (i.e. $\displaystyle e_0(R)=S$) and that $\displaystyle \text{ker}e_0$ is a maximal ideal of $\displaystyle R.$

Therefore the first question is : what is $\displaystyle \text{ker}e_0$ ?