What is an example for:
An extension of rings $k \subset R$ where $k$ is a finite field, $R$ is a finite dimensional vector space over $k$, $R$ is reduced, and $R \neq k[r]$ for all $r \in R$.
So, initially I thought that for a prime $p$, $k = \mathbb{F}_p$ and $R = \mathbb{F}_p \times \mathbb{F}_p$ would do the trick, but note that this does not satisfy the last condition as $\mathbb{F}_p \times \mathbb{F}_p = \mathbb{F}_p[(1,0)]$ for the idempotent $(1,0) \in \mathbb{F}_p \times \mathbb{F}_p$. This is true because for any $(a,b) \in \mathbb{F}_p \times \mathbb{F}_p, (a,b) = b(1,1) + (a-b)(1,0) \in \mathbb{F}_p[(1,0)]$. This is the example our professor seems to have had in mind too, so that now that it is seen to be incorrect, I am not sure whether an example exists.