# Atiyah Macdonald Exercise 5.22

This is a problem in Atiyah Macdonald, Commutative Algebra.

Problem 5.22 $$S$$ is a subring of an integral domain $$R$$. $$R$$ is a finitely generated $$S$$ algebra. If the Jacobson radical of $$S$$ is $$0$$, then the Jacobson radical of $$R$$ is $$0$$.

I followed the hint provided in the book as follows.

The purpose is to show for each element of $$R$$, there is a maximal ideal of $$R$$ avoiding it. Pick a $$0\neq v\in R$$. Since we have injective maps $$S\to R$$ and $$R\to R_v$$, we embed $$S$$ in $$R_v$$ as a subring and this $$R_v$$ is finitely generated as well over $$S$$.

It is clear that there is $$0\neq s\in A$$ such that we can extend the ring homomorphism $$\phi:S\to\Omega$$ to $$\tilde{\phi}:R_v\to\Omega$$ where $$\Omega$$ is some algebraically closed field. Here $$\Omega$$ is chosen to be the algebraically closed field of $$S/m$$ where $$s\not\in m$$ as Jacobson radical of $$S$$ is 0. It is clear that $$v$$ under the map $$R_v\to\Omega$$ is not zero or we will have $$1$$ being sent to $$0$$.

However, I am stuck at showing $$\ker(R_v\to\Omega)\subset R_v$$ is maximal ideal since this will lead to $$\ker(R_v\to\Omega)\cap B$$ is maximal. It is not even clear that this will be maximal. However, I can show if $$S$$'s Jacobson radical is trivial, then $$R=S[x]$$'s is trivial. It is possible that $$R=S[x]/Q$$ where $$Q$$ is some prime ideal and Jacobson radical of $$R$$ is the intersection of maximal ideals containing $$Q$$. I also noticed that quotient and localization do not necessarily commute with infinite intersection. What other options do I have here?

You have a non-trivial ring homomorphism $\tilde{\phi}: R_v \to \Omega$ with $\ker(\tilde{\phi}\mid S)=m$. Hence there are ring extensions $$k=S/m \le R_v/\ker(\tilde{\phi}) \le \Omega$$ Now, by the exercise below, $R_v/\ker(\tilde{\phi})$ is a field, i.e. $\ker(\tilde{\phi}) \le R_v$ is maximal.
 Exercise: Let $k$ be a field and let $\Omega$ be an algebraic extension of $k$. Then each subring $k \le A \le \Omega$ is a field.