Derivations of $k[x]/(x^2)$

Question

I am trying to find all the derivations of $k[x]/(x^2)$. Let us consider only outer (i.e. not inner) derivations. It is easy to see that the map given by $$ \delta: r + sx \mapsto sx $$ is a derivation, but how do we find all the outer derivations in general? Thanks for your help.

Answer 1

Let $A$ be your algebra and consider a linear map $f:A\to A$. As $\{1,x\}$ is a basis of $A$, there are scalars $a,b,c,d$ such that $f(1)=a+bx$ and $f(x)=c+dx$, and these four scalars complete determine the map $f$. Suppose now that $f$ is a derivation. Since every derivation maps $1$ to $0$, we must have $a=b=0$. Since $x^2=0$, the Leibniz rule tells us that $$0=f(xx)=xf(x)+f(x)x=x(c+dx)+(c+dx)x=2cx.$$ (Assuming that the characteristic of your field $k$ is not $2$), then we see that $c=0$.

It follows from this that every derivation of $A$ has $f(1)=0$ and $f(x)=dx=0$, for some $d\in k$. Every such map is a derivation, as you can easily check, so this describes all derivations.

The algebra is commutative, so there are no nonzero inner derivations.