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Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang; Feng Wei; Ajda Fošner — 2019

Czechoslovak Mathematical Journal

Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵 (𝒢)$ be the center of $𝒢$ . Suppose that $𝔮 : 𝒢 \times 𝒢 \to 𝒢$ is an $ℛ$ -bilinear mapping and that $𝔗_{𝔮} : 𝒢 \to 𝒢$ is a trace of $𝔮$ . The aim of this article is to describe the form of $𝔗_{𝔮}$ satisfying the centralizing condition $[𝔗_{𝔮} (x), x] \in 𝒵 (𝒢)$ (and commuting condition $[𝔗_{𝔮} (x), x] = 0$ ) for all $x \in 𝒢$ . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $𝔗_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

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