Let $A=\mathbb{F}_p[x,y]$, the commutative polynomial algebra on two variables over the finite field $\mathbb{F}_p$. Define a derivation on an algebra as a map which satisfies the Leibniz rule, ie if $d$ is a derivation then
$d(ab)=ad(b)+bd(a)$
Let $Der(A)$ be the $\mathbb{F}_p$-module (algebra?) of derivations of $A$. I'm interested in knowing what $Der(A)$ is. First of all, I'm not sure exactly how complicated the algebraic structure it carries is. I know that it's at least a $\mathbb{F}_p$-module. But is it an algebra? I suspect no, but I can't nail down a reason why. Is it a module over a bigger ring? $\mathbb{F}_p[x,y]$ seems like it might be a good choice- but again, I'm not very sure of this. Finally, what's an explicit description of elements of $Der(A)$? Is it just linear combinations of $\frac{d}{dx}$ and $\frac{d}{dy}$?
I apologize if this is a bit overly broad, but I'm encountering derivations over finite fields for the first time, and I'd like to really understand what's going on.