Posner's second theorem and annihilator conditions with generalized skew derivations
Volume 129 / 2012
Colloquium Mathematicum 129 (2012), 61-74
MSC: 16W25, 16N60.
DOI: 10.4064/cm129-1-4
Abstract
Let $\mathcal{R}$ be a prime ring of characteristic different from $2$, $\mathcal{Q}_r$ be its right Martindale quotient ring and $\mathcal{C}$ be its extended centroid. Suppose that $\mathcal{G}$ is a non-zero generalized skew derivation of $\mathcal{R}$ and $f(x_1, \ldots, x_n)$ is a non-central multilinear polynomial over $\mathcal{C}$ with $n$ non-commuting variables. If there exists a non-zero element $a$ of $\mathcal{R}$ such that $a[\mathcal{G}(f(r_1, \ldots, r_n)),f(r_1, \ldots, r_n)]=0$ for all $r_1, \ldots, r_n \in \mathcal{R}$, then one of the following holds:
(a) there exists $\lambda \in \mathcal{C}$ such that $\mathcal{G}(x)=\lambda x$ for all $x\in \mathcal{R};$
(b) there exist $q\in \mathcal{Q}_r$ and $\lambda \in \mathcal{C}$ such that $\mathcal{G}(x)=(q+\lambda)x+xq$ for all $x\in \mathcal{R}$ and $f(x_1, \ldots, x_n)^2$ is central-valued on $\mathcal{R}$.
