2 BERNARD R. McDONALD

in 1977.

There has been considerable interest in this problem. In addition to the

above-noted survey by Marcus [10] of this and related problems, there is an

earlier survey by Marcus [9] and, more recently, a survey in the Ph.D. thesis

of Robert Grone [3] which lists 103 related papers on linear mapping problems

over fields. Even here, Grone fails to list the extensive literature

concerning the automorphisms of the classical linear groups which is also

relevant to these problems.

In 1980 D. G. James [8] classified the linear mappings of (R) which

preserved determinant where R was in integral domain. It was this paper

that initiated our interest in this problem for the case where R is a

commutative ring; and, in Section (H) we give the solutions of determinant

preservers for an arbitrary commutative ring. W. Waterhouse [21] also began

to work on this problem at that time and has communicated to me a separate

solution of the determinant preservers over a commutative ring by the use of

group scheme techniques. Many problems of the above type are related to the

study of an affine group scheme and Waterhouse in [21] provides an excellent

exposition of this point of view.

On the other hand, the classification of rank one preservers has also

been extended to division rings by W. J. Wong [22]. Thus, there exist analogs

for noncommutative rings.

In this paper, we extend the theory to an arbitrary commutative ring.

The approach is to adopt the thesis of Marcus and attack initially the problem

of rank one preservers. When this is complete, we deduce from it the form of

the determinant preservers and several other invariant preserving linear

mappings.

The outline of the paper is as follows:

In Section (B) we discuss the decomposition of (R) when idempotents

are present in R . This is due to the transpose mapping which is significant

throughout this discussion; however, for a splitting of R into two subrings