ABSTRACT

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

Classification

15A33, 15A72, 13C10

Key Words

Matrices over commutative rings,

Invariant preserving maps.

Library of Congress Cataloging in Publication Data

McDonald, Bernard R.

R-linear endomorphisms of (R) [subscript n]

preserving invariants.

(Memoirs of the American Mathematical Society, ISSN

0065-9266 ; no. 287)

"November 1983."

Bibliography: p.

1. Matrix rings. 2. Commutative rings.

3. Invariants. I. Title. II.Series.

QA3.A57 no. 287 [QA188] 510s [512'.5] 83-15648

ISBN 0-8218-2287-X

IV