Automorphisms of $\textrm {GL}_{n}(R)$

Abstract:

Let R denote a commutative ring having 2 a unit. Let ${\text {G}}{{\text {L}}_n}\left ( R \right )$ denote the general linear group of all $n \times n$ invertible matrices over R. Let $\wedge$ be an automorphism of ${\text {G}}{{\text {L}}_n}\left ( R \right )$. An automorphism $\wedge$ is “stable” if it behaves properly relative to families of commuting involutions (see §IV). We show that if R is connected, i.e., 0 and 1 are only idempotents, then all automorphisms $\wedge$ are stable. Further, if $n \geqslant 3$, R is an arbitrary commutative ring with 2 a unit, and $\wedge$ is a stable automorphism, then we obtain a description of $\wedge$ as a composition of standard automorphisms.