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
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
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