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Derivations with Invertible Values

Published online by Cambridge University Press:  20 November 2018

Jeffrey Bergen ,
I. N. Herstein  and
Charles Lanski
Jeffrey Bergen
Affiliation:
De Paul University Chicago, Illinois
I. N. Herstein
Affiliation:
University of Chicago, Chicago, Illinois
Charles Lanski
Affiliation:
University of Southern California, Los Angeles, California
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In this paper we study a question which, although somewhat special, has the virtue that its answer can be given in a very precise, definitive, and succinct way. It shows that the structure of a ring is very tightly determined by the imposition of a special behavior on one of its derivations.

The problem which we shall examine is: Suppose that R is a ring with unit element, 1, and that d ≠ 0 is a derivation of R such that for every x ∊ R, d(x) = 0 or d(x) is invertible in R; must R then have a very special structure?

As we shall see, the answer to this question is yes, in particular we show that except for a special case which occurs when 2R = 0, R must be a division ring D or the ring D2 of 2 × 2 matrices over a division ring.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 2 , 01 April 1983 , pp. 300 - 310
Copyright
Copyright © Canadian Mathematical Society 1983