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MATHEMATICA BOHEMICA, Vol. 133, No. 2, pp. 197-208 (2008)
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Free actions on semiprime rings

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Muhammad Anwar Chaudhry, Mohammad S. Samman

* 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 (MSC2000):** 16N60

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