Maps determined by rank-\(s\) matrices for relatively small \(s\)
type: Item
instance of: scholarly article
title: Maps determined by rank-\(s\) matrices for relatively small \(s\)
zbMATH Open document ID: 1367.15002
author: Jianwei Zhu
author: Chen Li
author: Xiaowei Xu
published in: Aequationes Mathematicae
publication date: 2017-04-28T00:00:00Z
review text: 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 \((m-1)\).
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 6708789
zbMATH Keywords: additive maps
zbMATH Keywords: rank-\(s\) matrices
zbMATH Keywords: nonadditive subsets
reviewed by: Rabe von Randow
MaRDI profile type: Publication
full work available at URL: s00010-017-0469-8
OpenAlex ID: W2587626554
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DOI: 10.1007/S00010-017-0469-8
description: scientific article; zbMATH DE number 6708789