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Let A be a triangular algebra. The problem of describing the form of a bilinear map B :for all x ∈ A is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.  2004 Elsevier Inc. All rights reserved.

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We consider Jordan derivations of a unital algebra A having a nontrivial idempotent. It turns out that on unital algebras there exist Jordan derivations that are not derivations. For this purpose we introduce the term a singular Jordan derivation, which is a proper Jordan derivation of the form that depends on Peirce decomposition of the unital algebra A. Singular Jordan derivations are usually antiderivations. The main result of the paper states that under certain conditions every Jordan derivation of A is the sum of a derivation and a singular Jordan derivation.

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