Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Let A be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S:A⟶A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J. We show that, under mild conditions, every generalized Jordan n-derivation S:A⟶A is of the form Sx=λx+Jx in the current work. As an application, we give a description of generalized Jordan derivations for the condition n=2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.