Jordan Left {g, h}-Derivation over Some Algebras

Abstract

In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over $${{\cal M}_n}\left(C \right)$$ and ℍℝ, for g ≠ h. Examples are given which show that every Jordan left {g, h}-derivation over $${{\cal T}_n}\left(C \right)$$ , $${{\cal M}_n}\left(C \right)$$ and ℍℝ are not left {g, h}-derivations. Also, the Jordan left {g, h}-derivations over $${{\cal T}_n}\left(C \right)$$ , $${{\cal M}_n}\left(C \right)$$ and ℍℝ are right centralizers, where C is a 2-torsionfree commutative ring. Moreover, we prove the result of Jordan left {g, h}-derivation to be a left {g, h}-derivation over tensor products of algebras as well as for algebra of polynomials.