Commuting maps on some subsets of matrices that are not closed under addition
Let \(M_{n}(\mathbb K)\) be the ring of all \(n\times n\) matrices over a field \(\mathbb K\). The author describes all additive mappings \(G:M_{n}(\mathbb K)\to M_{n}(\mathbb K)\) satisfying the condition \(G(x)x=xG(x)\) for all \(x\) from one of the following subsets of \(M_{n}(\mathbb K)\): (i) the set of all singular \(x\in M_{n}(\mathbb K)\), and (ii) the set of all invertible \(x\in M_{n}(\mathbb K)\); in the second case the extra assumption \(|\mathbb K|>2\) is made. In both cases such a \(G\) has the form \(x\mapsto\lambda x + \mu(x)\), where \(\lambda\) is a scalar multiple of the identity matrix and \(\mu\) is an additive mapping of \(M_{n}(\mathbb K)\) into its centre. An illustrative example shows that in (ii) the assumption on \(|\mathbb K|\) must not be dropped.
scientific article; zbMATH DE number 6034537