Is a finitely generate $k$-algebra that has no nilpotent element ($k$ is a field) an integral domain?
Solution 1:
Not necessarily. Consider the union of the $x$ and $y$ axes in $\Bbb{A}^2_k$. It is the vanishing locus of $f(x,y)=xy$. The ring $k[x,y]/(xy)$ is not an integral domain because $\overline{x}\cdot \overline{y}=0$. However, being the coordinate ring of this affine algebraic set it is reduced (which is not hard to check directly, as well). However, if you require that your algebraic set is irreducible in addition, you will find that its coordinate ring is an integral domain.