(jordan)centralizers

begin If $F,D:Rto R$ are additive mappings which satisfy $F(x^{n}y^{n})=x^nF(y^{n})+y^nD(x^{n})$ for all $x,yin R$. Then, $F$ is a generalized left derivation with associated Jordan left derivation $D$ on $R$. Similar type of result has been done for the other identity forcing to generalized derivation and at last an example has given in support of the theorems.the formula is not displayed correctly!

The main purpose of this article is to prove the following main result: Let R be a 2-torsion free semiprime ring and T : R → R be a Jordan left centralizer associated with an l-semi Hochschild 2-cocycle α: R ⨯ R → R. Then, T is a left centralizer associated with α. In order to show application of this result, several corollaries concerning Jordan generalized derivations, Jordan σ-derivations, Jordan generalized σ-derivations and Jordan (σ, τ )-derivations will be presented.