Pull back image of maximal ideal under surjective ring homomorphism is maximal

Question

Let $f :R \to S$ be a surjective ring homomorphism , $M$ be a maximal ideal of $S$ , I am writing a proof showing $f^{-1}(M)$ is a maximal ideal of $R$ , Please verify whether it is correct or not .

Proof :- Let $J$ be an ideal such that $f^{-1}(M) \subset J \subseteq R$ , we want to show $J=R$ i.e. $R \subseteq J$. Now as $f$ is surjective , $M=f(f^{-1}(M)) \subseteq f(J) \subseteq S$ . As $f$ is surjective , $f(J)$ is an ideal of $S$ . Now if it were possible that $M=f(J)$ , then $x \in J \implies f(x) \in f(J)=M \implies x \in f^{-1}(M)$ , so $J \subseteq f^{-1} (M)$ ,contarry to our assumption $f^{-1}(M) \subset J$ . Thus $M \ne f(J)$ , $f(J)$ is an ideal of $S$ containing $M$ ; since $M$ is maximal ideal in $S$ , so $f(J)=S$ . Then $x \in R \implies f(x) \in S=f(J) \implies \exists j \in J : f(x)=f(j) \implies f(x-j)=0_S \in M $

$\implies x-j \in f^{-1} (M) \subset J \implies x=x-j+j \in J$ , so $R \subseteq J$

Is the proof correct . Please comment .

Answer 1

Your proof does indeed seem to be correct. It is written in a very convoluted way, however. Perhaps you should start with:

Let $J$ be an ideal containing $f^{-1}(M)$. As $f$ is surjective, $f(J)$ is an ideal, and it contains $M$. As $M$ is maximal, either $f(J)=M$ or $f(J)=S$. If $f(J)=M$, then...

Notice that this avoids your imprecision with the "contrary with our assumption", which actually is not contrary to any assumption you made. Also, it cleans up the proof, by making the chain of deduction clearer.

If I may suggest, a cleaner way of proving this is by an altogether different method, bypassing elements. Since $M$ is maximal, $k:=S/M$ is a field. Since $f$ is surjective, its composition with the canonical projection $\bar{f}:R\to k$ is a surjection. This means that $\ker \bar{f}$ is a maximal ideal. Can you compute it?

Answer 2

I think it would be better to deduce this from the more general correspondence theorem.

This says that there is a one-to-one, inclusion-preserving correspondence between the ideals of $S$ and the ideals of $R$ which contain the kernel $K$ of $f$.

Answer 3

If $f\colon R\to S$ is surjective, then for any ideal $J\subset S$, the induced ring homomorphism $$\varphi\colon R/f^{-1}J\to S/J$$ is also surjective. In fact, the kernel of $\varphi$ is trivial, so $\varphi$ is an isomorphism. Recall that an ideal $J$ of a ring $S$ is maximal if and only if $S/J$ is a field. So, $J\subset S$ is maximal if and only if $S/J$ is a field if and only if $R/f^{-1}J$ is a field if and only if $f^{-1}J\subset R$ is maximal.