Muhammad Anwar Chaudhry, Bahauddin Zakariya University, Center for Advanced Studies in Pure and Applied Mathematics, Multan, Pakistan, e-mail: chaudhry@bzu.edu.pk; Mohammad S. Samman, King Fahd University of Petroleum & Minerals, Department of Mathematical Sciences, Dhahran 31261, Saudi Arabia, e-mail: msamman@kfupm.edu.sa
Abstract: 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.
Keywords: prime ring, semiprime ring, dependent element, free action, centralizer, derivation
Classification (MSC 2000): 16N60
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