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Keywords:
prime ring; semiprime ring; dependent element; free action; centralizer; derivation
Summary:
We identify some situations where mappings related to left centralizers, derivations and generalized $(\alpha ,\beta )$-derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation $T$, of a semiprime ring $R$ the mapping $\psi \: R \rightarrow R$ defined by $\psi (x)=T(x) x - x T(x)$ for all $x \in R$ is a free action. We also show that for a generalized $(\alpha , \beta )$-derivation $F$ of a semiprime ring $R,$ with associated $(\alpha , \beta )$-derivation $d,$ a dependent element $a$ of $F$ is also a dependent element of $\alpha + d.$ Furthermore, we prove that for a centralizer $f$ and a derivation $d$ of a semiprime ring $R$, $\psi = d\circ f$ is a free action.
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