Centralizing maps in prime rings with involution
Let \(R\) be a ring with center \(Z\). For \(x, y \in R\), denote \([x,y] = xy - yx\). An additive map \(f : K \to K\) defined on a subset \(K \subset R\) is called centralizing if \([f(x), x] \in Z\) for all \(x\in K\). Centralizing maps have been studied by a number of authors, starting with Posner in 1957. The present paper focuses on the case where \(R\) is a prime ring of characteristic different from 2 and \(K\) is the group of skew-symmetric elements in \(R\) under some involution \(*\). The main result is a description of all the centralizing maps on \(K\), under the hypotheses that \([K,K] \neq \{0\}\) and that \(R\) is not an order in a 16-dimensional central simple algebra. It is shown that every such map \(f\) has the form \(f(x) = \lambda x + \mu(x)\) for some \(\lambda\) in the extended centroid \(C\) of \(R\) and some additive map \(\mu : K \to C\). In particular, the map \(f\) is actually commuting, in the sense that \([f(x),x] = 0\) for all \(x \in K\).
scientific article; zbMATH DE number 554246