Question

This question is a follow up question to this question. So my question is:

For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of Theorem A and B in O'Meara The automorphisms of the linear groups over any integral domain is that this is the case (for $n\ge 3$) for any integral domain, whose underlying additive abelian group is finitely generated.

However this is just a computation and I am wondering, whether this question has already been studied somewhere more systematically or if there are other results that also have such a corollary.

Answer 1

This question has been studied even for non-commutative associative rings. See, for example Golubchik, I. Z.; Mikhalëv, A. V. Isomorphisms of the general linear group over an associative ring. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 3, 61–72. They prove that if the ring $R$ contains $1/2$ (that is $2$ is invertible), then every isomorphism $\phi: GL_n(R)\to GL_n(R)$, $n\ge 3$, is standard on the subgroup $GE_n(R)$ generated by the elementary and diagonal matrices. "Standard" means "generated by an automorphism of $R$.