The Stacks project

Exercise 111.16.3. Let $k$ be a field. Please use Remark 111.16.2.

1. Let $R$ be a $k$-algebra. Suppose that $\dim _ k R < \infty$ and that $R$ is a domain. Show that $R$ is a field.

2. Suppose that $R$ is a finitely generated $k$-algebra, and $f\in R$ not nilpotent. Show that there exists a maximal ideal ${\mathfrak m} \subset R$ with $f\not\in {\mathfrak m}$.

3. Show by an example that this statement fails when $R$ is not of finite type over a field.

4. Show that any radical ideal $I \subset {\mathbf C}[x_1, \ldots , x_ n]$ is the intersection of the maximal ideals containing it.