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)?