Abstract. Biadditive mappings B: R x R -> R where R is a prime ring with certain additional properties, satisfying B(x, x)x = xB(x, x) for all x £ R, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings.

Abstract. A linear map T from a Banach algebra A into another B preserves zero products if T (a)T (b) = 0 whenever a, b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T : A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras.Our method involves continuous bilinear maps φ : A × A → X (for some Banach space X) with the property that φ(a, b) = 0 whenever a, b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies φ(aµ, b) = φ(a, µb) for all a, b ∈ A and for all µ from the closure with respect to the strong operator topology of the subalgebra of M(A) (the multiplier algebra of A) generated by doubly power-bounded elements of M(A). This method is also shown to be useful for characterizing derivations through the zero products.Introduction. S. Banach [4] was the first to describe isometries on L p ([0, 1]) with p = 2. Although Banach did not give the full proof for this case (this was provided by J. Lamperti [26]), he made the key observation that isometries on L p ([0, 1]) must take functions with disjoint support into functions with disjoint support. This property arises in a variety of situations and has been considered by several authors. For example, in the theory of Banach lattices there is an extensive literature about linear maps T : X → Y , where X and Y are Banach lattices, with the property that |T (x)| ∧ |T (y)| = 0 whenever x, y ∈ X are such that |x| ∧ |y| = 0. Such maps are usually called disjointness preserving operators or d-homomorphisms. The reader interested in this setting is referred to the monograph [1]. The concept of a disjointness preserving operator was exported to function algebras by E. Beckenstein and L. Narici (see [5] for general information).

Introduction.A well-known theorem of E. Posner [10] to the set of all generalized derivations of A. We consider the case when A is an ultraprime normed algebra (Theorem 1), the case when d x = d 2 and A is ultrasemiprime (Theorem 2), and finally, the case when A is a von Neumann algebra (Theorem 3). As a consequence of these results we obtain a partial answer to Mathieu's question [8]: what is the norm of the composition of two derivations in a prime C*-algebra?Our results will follow easily from two entirely elementary observations; it is our aim in this paper to point out the kind of method that could be used.