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Two-sided zero product determined algebras. (English) Zbl 1500.16039

Summary: An algebra \(A\) is said to be two-sided zero product determined if every bilinear functional \(\varphi : A \times A \to F\) satisfying \(\varphi(x, y) = 0\) whenever \(x y = y x = 0\) is of the form \(\varphi(x, y) = \tau_1(x y) + \tau_2(y x)\) for some linear functionals \(\tau_1, \tau_2\) on \(A\). We present some basic properties and equivalent definitions, examine connections with some properties of derivations, and as the main result prove that a finite-dimensional simple algebra that is not a division algebra is two-sided zero product determined if and only if it is separable.

MSC:

16W25 Derivations, actions of Lie algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16P10 Finite rings and finite-dimensional associative algebras
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