On α−Centralizer Mappings in Semiprime Rings.

Let R be a 2−torsion free semiprime ring, α an automorphism and U a noncentral square-closed Lie ideal of R. An additive mapping T : R → R is called a left (resp. right) α−centralizer of R if T(xy) = T(x)α(y) (resp. T(xy) = α(x)T(y)) holds for all x, y ∈ R. In this paper, we proved the following result: i) If T (uvu) = α (u) T (v) α (u) for all u, v ∈ U, then T is a left α−centralizer, ii) If T(uvu) = T(u)α(vu) (resp. T(uvu) = α(uv)T(u)) for all u, v ∈ U, then T is a left (resp. right) α-centralizer on U. [ABSTRACT FROM AUTHOR]