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 [6]inhisclassification

and discussion ofthe R-algebra automorphisms of (R) in 1980. His work

was a reformulation ofearlier andmore general work byRosenberg andZelinsky

[19] 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

(R)n©(R)n.

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