An investigation of anticommutators in some algebras

A natural analogue of (additive) commutators in an associative algebra [Formula: see text] of characteristic different from [Formula: see text] is the anticommutator, defined by [Formula: see text] for elements [Formula: see text]. Although the anticommutator is used less frequently, it appears in the definition of Clifford algebras and Jordan algebras, as well as in the derivation of the Dirac equation in particle physics. If [Formula: see text] is unital with identity element [Formula: see text], then any element [Formula: see text] is expressed as [Formula: see text]. With this phenomenon in place, this paper considers the question of whether every element [Formula: see text] can be expressed [Formula: see text] for some noncommuting elements [Formula: see text], meaning that their commutator [Formula: see text] is nonzero. Inspired by Terence Tao’s work [J. Oper. Theory 82, No. 2, 369-382 (2019)], we also examine what happens when [Formula: see text] is close to [Formula: see text].