On invertible bimodules and automorphisms of noncommutative rings

In this article, we attempt to generalize the result that for a commu- tative ring R the outer automorphism group of i?-automorphisms of M"(R) is abelian of exponent n . It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive di- mension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring. Let R be a ring with 1. In this article we are interested in comparing the automorphism group of Mn{R), the ring of « x « matrices over R, and the automorphism group of R. In particular, if R is commutative, it is a classical result that the outer automorphism group of .R-automorphisms of Mn{R) is an abelian group of exponent dividing «. This can be restated in terms of invertible bimodules. The result is equivalent to the easy fact that any projective rank one module X with nX free over a commutative ring has order dividing « in the Picent group of R—see (RZ and KO). In the noncommutative case, there is still a well known relationship between automorphisms and the Picard