Maps determined by rank-\(s\) matrices for relatively small \(s\)
Let \(M\) be the ring of all \(n \times n\) matrices over a field \(\mathbb{K}\). Let \(s\) be an integer with \(1 \leq s < \frac{n}{2}\) and \(m\) the least integer \(\geq \frac{n}{s}\). A map \(g: M\to M\) is said to satisfy condition (*) if \(g(\sum_{1}^{m}A_i)=\sum_{1}^{m}g(A_i)\) for any \(m\) rank-\(s\) matrices \(A_i\). Then the following is clear: if \(\mathrm{char}\mathbb{K}\) divides \((m-1)\), then the map \(g(x) = f(x) + A\) with \(x \in M\), \(f\) additive and \(A\neq0\), satisfies condition (*) but is not additive. A natural problem is whether or not a nonadditive \(g\) satisfying condition (*) can take any other form than that given above. In this paper, the authors show that the answer is negative, and that \(g\) is additive if \(\mathrm{char}\mathbb{K}\) does not divide \((m-1)\).
scientific article; zbMATH DE number 6708789
scientific article