Two-sided zero product properties on symmetric algebras


KOŞAN M. T., Lee T., Lin J.

LINEAR ALGEBRA AND ITS APPLICATIONS, vol.655, pp.186-201, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 655
  • Publication Date: 2022
  • Doi Number: 10.1016/j.laa.2022.09.014
  • Journal Name: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, MathSciNet, zbMATH
  • Page Numbers: pp.186-201
  • Keywords: Symmetric algebra, Separable algebra, Bilinear functional, Derivation, (Two-sided) zero product determined algebra, Jacobson radical, DERIVATIONS, XY
  • Gazi University Affiliated: Yes

Abstract

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.