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 $\psi R\to R$ defined by $\psi \left(x\right)=T\left(x\right)x-xT\left(x\right)$ 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...