INVARIANT PRESERVING LINEAR HAPS 3
induced byan idempotent, the"transpose" mayaffect only onesummand of the
induced splitting of (R) andleave the other summand fixed. Thus, we have
many "transposes", andthey form a group which maybe identified with the set
of idempotents of R .
In Section (C) we summarize the standard theory of equivalence
transformations of (R) .
The heart ofthis paper is Section (D). Here we develop thetheory of
invertible R-submodules of (R) © (R) . The work ispatterned afterthe
development of thetheory for (R) byM. Issacs inhisclassification
and discussion ofthe R-algebra automorphisms of (R) in 1980. His work
was a reformulation ofearlier andmore general work byRosenberg andZelinsky
 in 1961. Here we identify "generalized" equivalence transformations of
(R) certain "twisted" lines, i.e., rank one projectives, in "twisted"
planes, i.e., direct sums oftworank oneprojectives, which sit in
In Section (E) we generalize the concept of a rank one matrix over a
field . Here we say that a matrix A in (R) has rank one i f it s range,
as a linear mapping, is a rank one projective R-module; equivalently, i f the
columns of A generate a rank one projective. In this section, we develop
the theory of these rank one mappings.
In Section (F), we prove the Marcus-Moyls Theorem for a local commutative
ring R . Then, in Section (G), we develop the general case for an arbitrary
commutative ring by the use of localization techniques from commutative
algebra. In this section we complete the classification of rank one preservers
and our results are summarized in the paper's main theorem - Theorem (G.4).
Section (H) concerns applications of Theorem (G.4) to other invariant
preserving questions. In particular, the determinant preserving linear maps
are classified by Theorem (H.4), thus giving a modern formulation of
Frobenius' original theorem.
I t remains to express appreciation to those who contributed in some