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Derivations with invertible values on a multilinear polynomial


Author: Tsiu Kwen Lee
Journal: Proc. Amer. Math. Soc. 119 (1993), 1077-1083
MSC: Primary 16W25; Secondary 16K40, 16N60, 16R99
MathSciNet review: 1156472
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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$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1156472-7
Keywords: Semiprime rings, derivations, central polynomials, differential identities
Article copyright: © Copyright 1993 American Mathematical Society