Traces of multiadditive maps on rank-s matrices

Abstract

Let m, n be integers such that 1<m<n. Let $\mathcal{R}=M_n(\mathbb{D})$ be the ring of all $n\times n$ matrices over a division ring $\mathbb{D}$, $\mathcal{M}$ an additive subgroup of $\mathcal{R}$ and $G:{\mathcal{R}}^m\to \mathcal{R}$ an m-additive map. In this paper, under a mild technical assumption, we prove that ${\delta }_1(x)=G(x,\dots ,x)\in \mathcal{M}$ for each rank-s matrix $x\in \mathcal{R}$ implies ${\delta }_1(x)\in \mathcal{M}$ for each $x\in \mathcal{R}$, where s is a fixed integer such that $m\le s<n$, which has been considered for the case s = n in [Xu X, Zhu J., Central traces of multiadditive maps on invertible matrices, Linear Multilinear Algebra 2018; 66:1442–1448]. Also, an example is provided showing that the conclusion will not be true if s<m. As applications, we also extend the conclusions by Liu, Franca et al., Lee et al. and Beidar et al., respectively, to the case of rank-s matrices for $m\le s<n$.

Keywords:

• multiadditive maps
• traces
• rank-s matrices
• commuting maps
• Engel condition

AMS classifications:

• 16W25
• 15A03
• 15A23

Acknowledgments

We would like to express our sincere thanks to the referee for correcting punctuation mistakes and indicating the direct implication from Theorem 3.4 (1) to Corollary 3.2, which help us simplify the proof of Corollary 3.2. In particular, the referee's idea on the proof of Lemma 2.1 leads to its current version, which is clearer than its previous form.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Xiaowei Xu is supported by the National Natural Science Foundation of China [grant number 11971289]; Haoran Yu is supported by the National Natural Science Foundation of China [grant numbers 12001225 and 12171194].