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LINEAR ALGEBRA AND ITS APPLICATIONS, vol.655, pp.186-201, 2022 (SCI-Expanded)
We characterize bilinear functionals phi on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., phi(x, y) = 0 whenever xy = yx = 0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x, y) bar right arrow tau(1()xy) + tau(2)(yx) for x, y is an element of A, where tau(1), tau(2) are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A, there exists a is an element of A such that, for all x is an element of A, d(x) - [a, x] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd. (C) 2022 Elsevier Inc. All rights reserved.