On maps preserving products of matrices (Q1713305)

Let \(R=M_n(D)\), where \(n\ge 2\) and \(D\) is a division ring with characteristic different from \(2\), let \(Z\) be the center of \(R\), and let \(m\) and \(k\) be (fixed) invertible elements in \(R\). It is shown that if \(f:R\to R\) is a bijective additive map satisfying \(f(x)f(y) = m\) whenever \(x,y\in R\) are such that \( xy = k\), then there exists an automorphism or an antiautomorphism \(\varphi\) of \(R\) such that \(f(x) = f(1)\varphi(x)\) for all \(x \in R\). In the case where \(D=\mathbb C\), the authors also consider a similar condition in which the role of the ordinary product is replaced by the Jordan product \(x\circ y=xy+yx\).