Jordan alpha-centralizers in rings and some applications - Claus Haetinger; Shakir Ali (Boletim da Sociedade Paranaense de Matemática, 2008)
Let R be a ring, and alpha be an endomorphism of R. An additivemapping H: R ightarrow R is called a left alpha-centralizer (resp. Jordan left alpha-centralizer) if H(xy) = H(x)alpha(y) for all x; y in R (resp. H(x^2) = H(x)alpha(x) for all x in R). The purpose of this paper is to prove two results concerning Jordan alpha-centralizers and one result related to generalized Jordan (alpha; eta)-derivations. The result which we referstate as follows: Let R be a 2-torsion-free semiprime ring, and alpha be an automorphism of R. If H: R ightarrow R is an additive mapping such that H(x^2) = H(x)alpha(x) for every x 2 R or H(xyx) = H(x)alpha(yx) for all x; y in R, then H is a left alpha-centralizer on R. Secondly, this result is used to prove that every generalized Jordan (alpha; eta)-derivation on a 2-torsion-free semiprime ring is a generalized (alpha; eta)-derivation. Finally, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the various theorems were not superfluous.