Derivations with invertible values on a multilinear polynomial

Abstract:

Let $R$ be a semiprime $K$-algebra with unity, $d$ a nonzero derivation of $R$, and $f({x_1}, \ldots ,{x_t})$ a monic multilinear polynomial over $K$ such that $d(f({a_1}, \ldots ,{a_t})) \ne 0$ for some ${a_1}, \ldots ,{a_t} \in R$. It is shown that if for every ${r_1}, \ldots ,{r_t}$ in $R$ either $d(f({r_1}, \ldots ,{r_t})) = 0$ or $d(f({r_1}, \ldots ,{r_t}))$ is invertible in $R$, then $R$ is either a division ring $D$ or ${M_2}(D)$, the ring of $2 \times 2$ matrices over $D$, unless $f({x_1}, \ldots ,{x_t})$ is a central polynomial for $R$. Moreover, if $R = {M_2}(D)$, where $2R \ne 0$ and $f({x_1}, \ldots ,{x_t})$ is not a central polynomial for $D$, then $d$ is an inner derivation of $R$.