AUTOMORPHISMS OF GLn ( R ) BY BERNARD

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair (.R, S) is said to be stable if the ring of fractions of R, S R, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair (R, S) assume 2 is a unit in R and V is a free R-module of dimension > 3. This paper examines the action of a group automorphism of GL(V) (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of GL(V). (I). Introduction. Let F be a ring and GLn(R) the general linear group of « by « invertible matrices over F. If A is a group automorphism of GLn(R) then a basic problem is that of obtaining a description of the action of A on elements of GLn(R). First, what are the standard automorphisms? If o: F -*■ F is a ring automorphism then clearly a induces an isomorphism of the « by « matrix ring (R)n -*■ (R)n and since a unit maps to a unit we obtain an automorphism GLn(R) —> GLn(R) where A -* A". For A in GLn(R) we also have A -* A* where A* = (A*)~x (transpose-inverse). Each Q in GLn(R) provides an inner-automorphism A -* QAQ~X. Finally, for suitable group morphisms x: GLn(R) -*• center(GF„(F)) then A -*■ x(A)A is a group automorphism. Precisely, if A is an automorphism of GLn(R) then does A(A) = x(A)QAaQ~x for alM in GLn(R)