Let R be a commutative ring and (R) denote the n * n matrix ring
over R . In this paper we classify the R-linear mappings T : (R) -* (R)
which preserve rank one matrices. This classificatio n gives as a corollary
those R-linear mappings which preserve the determinant. Other invariant
preserving maps are also determined. These maps are invertibl e and we describe
the groups that they generate.
1980 Mathematics Subject
15A33, 15A72, 13C10
Matrices over commutative rings,
Invariant preserving maps.
Library of Congress Cataloging in Publication Data
McDonald, Bernard R.
R-linear endomorphisms of (R) [subscript n]
(Memoirs of the American Mathematical Society, ISSN
0065-9266 ; no. 287)
1. Matrix rings. 2. Commutative rings.
3. Invariants. I. Title. II.Series.
QA3.A57 no. 287 [QA188] 510s [512'.5] 83-15648