Posner's second theorem, multilinear polynomials and vanishing derivations

J. Aust. Math. Soc. 76 (2004), 357-368

Posner's second theorem, multilinear polynomials and vanishing derivations

Vincenzo De Filippis
Dipartimento di Matematica
Universitá di Messina
Salita Sperone 31
98166 Messina
Italia
enzo@dipmat.unime.it

and

Onofrio Mario Di Vincenzo
Dipartimento di Matematica
Universitá di Bari
Via Orabona 4
70125 Bari
Italia
divincenzo@dm.uniba.it

Abstract

Let

be a commutative ring with unity,

a prime K-algebra of characteristic different from 2,

and $\delta$ non-zero derivations of

, $f(x_1,\dots,x_n)$ a multilinear polynomial over

. If

$\delta ([d(f(r_1,\dots,r_n)),f(r_1,\dots,r_n)])=0 \quad\text{for all }\ r_1,\dots,r_n \in R,$

then $f(x_1,\dots,x_n)$ is central-valued on