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 define the lower socle of a semiprime algebra A as the sum of all minimal left ideals Ae where e is a minimal idempotent such that the division algebra eAe is finite dimensional. We study the connection between the condition that the elements a k , b k , 1 k n, lie in the lower socle of A and the condition that the elementary operator x 7 ! a 1 xb 1 þ Á Á Á þ a n xb n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.

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In this paper we prove the following result: Let R be a 2-torsion free semiprime ring. Suppose there exists an additive mapping T : R -> R such that T(xyx) = T(x)yx -xT(y)x + xyT(x) holds for all pairs x,y e R. Then T is of the form 2T(x) = qx + xq, where q is a fixed element in the symmetric Martindale ring of quotients of R.