Abstract
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 S x = λ x + J x 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.
Highlights
Xinfeng LiangLet us assume that A has an idempotent e ≠ 0, 1 and let f 1 − e
Where eAe and fAf are subalgebras with unitary elements e and f, respectively, eAf is an-bimodule and fAe is an-bimodule
It is worth to mention that A is isomorphic to a generalized matrix algebra [1]
Summary
Let us assume that A has an idempotent e ≠ 0, 1 and let f 1 − e In this case, A can be represented in the so called Peirce decomposition form. We assume that A satisfies exe.eAf 0 fAe.exe implies exe 0, (2). Some special examples of unital algebras with a nontrivial idempotents having the property (♣) are triangular algebras, matrix algebras, and prime (and in particular simple) algebras with nontrivial idempotent, nest algebras, standard operator algebras (see [2] for more details). It follows from (♣) that at least one of the bimodules eAf and fAe is nonzero.
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