Derivation of formal power series

Question

The basic idea of this question is to see if there is any other derivations than 'formal derivations'.

Let $\mathbb{K}$ be a field. Given a commutative $\mathbb{K}$-algebra $A$, a derivation of $A$ is a $\mathbb{K}$-linear map $D:A\rightarrow A$ satisfying $D(ab)=D(a)b+ aD(b)$. Consider the case when $\mathbb{K}=\mathbb{R}$ and the algebra $A$ is the formal power series $\mathbb{R}[[x]]$. One derivation of $\mathbb{R}[[x]]$ is $p(x)\frac{\partial}{\partial x}$, defined formally as $$ p(x) \frac{\partial}{\partial x} \left(\sum_{n=0}^\infty c_n x^n \right)=p(x)\cdot \left(\sum_{n=0}^\infty n\cdot c_n x^{n-1}\right)$$ for any $p(x)\in \mathbb{R}[[x]]$. My question is that, are there any derivations that is not of the form $p(x)\frac{\partial}{\partial x}$ as above? If yes, is there any reference/proof/example for existence of such derivations?

This question arose when I was thinking of vector fields of a $\mathbb{Z}$-graded manifold as a derivation of smooth functions. With my notion of graded manifolds, smooth functions are formal power series of virtual coordinates.

When I was thinking about it, since $\mathbb{R}[[x]]$ is not generated by $\{x\}$ as an algebra, there should be some other derivations but I could not make it clear. Any help to figure it out would be greatly appreciated.

Answer 1

Every $\mathbb{K}$-derivation $D$ of $\mathbb{K}[[x_1,\dots,x_n]]$ has the standard form $$D(f)=\sum_{i=1}^n p_i \frac{\partial f}{\partial x_i},$$ where of course $p_i=D(x_i)$.
Indeed, let $\mathfrak{m}$ be the maximal ideal of $\mathbb{K}[[x_1,\dots,x_n]]$: then clearly $D(\mathfrak{m}^{N+1})\subset \mathfrak{m}^N$ for all $N\geq0$, so $D$ is continuous for the $\mathfrak{m}$-adic topology. The claim follows because the formula holds when $f$ is a polynomial, $\mathbb{K}[x_1,\dots,x_n]$ is dense in $\mathbb{K}[[x_1,\dots,x_n]]$, and $\mathbb{K}[[x_1,\dots,x_n]]$ is Hausdorff.