We identify some situations where mappings related to left centralizers, derivations and generalized (α, β)-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 ψ: R → R defined by ψ(x) = T(x)x − xT(x) for all x ∈ R is a free action. We also show that for a generalized (α, β)-derivation F of a semiprime ring R, with associated (α, β)-derivation d, a dependent element a of F is also a dependent element of α + d. Furthermore, we prove that for a centralizer f and a derivation d of a semiprime ring R, ψ = d ◦ f is a free action.
For an arbitrary permutation σ in the semigroup Tn of full transformations on a set with n elements, the regular elements of the centralizer C(σ) of σ in Tn are characterized and criteria are given for C(σ) to be a regular semigroup, an inverse semigroup, and a completely regular semigroup.