Relation of the first Hochschild cohomology and the outer automorphism group

Question

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

(It is not true, by the two answers. Is it at least true that the Hochschild cohomology is zero if the outer automorphism group is finite?)

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this (at least for finite dimensional algebras)?

Answer 1

Another easy counter-example: take $X = \mathbb{N}$ as discrete topological space and $R = C(X, \mathbb{R})$ as continuous functions on it. These are just all functions. Equivalently, you can view them as smooth functions on the smooth manifold $X$. Now the first Hochschild cohomology is trivial since $X$ is zero-dimensional. Any automorphism is outer since $R$ is commutative. The pull-back with any bijection of $X$ gives an outer automorphism, quite many.

Answer 2

I don't believe this is true. Consider $D_{\mathbb{A}_{\mathbb{C}}^{1}}=:R$. Then we have $HH^{1}(R)=0$ by a standard computation. In fact $HH^{*}(D_{X})\cong H^{*}_{dR}(X)$ holds more generally. Now $Out(R)=Aut(R)$ as $R^{*}=\mathbb{C}^{*}$ is central. $Aut(R)$ includes all the transformations $$x\mapsto x+f(\partial), \partial\mapsto \partial,$$ where $f$ is any polynomial in one variable.