Let $Z$ be an algebra over a field $K$ such that $Z$ is generated by a single nilpotent element. Why is $Z$ a local ring?
Would this follow from the Chinese Remainder theorem?
Let $m$ be a maximal ideal of $Z$ then since $Z$ is Artinian then $Z \cong Z/m^{j}$ (by counting dimensions over $K$) for some natural $j$. But $Z/m^{j}$ is a local ring so we are done.