Abstract
Let m, n be integers such that 1<m<n. Let be the ring of all
matrices over a division ring
,
an additive subgroup of
and
an m-additive map. In this paper, under a mild technical assumption, we prove that
for each rank-s matrix
implies
for each
, where s is a fixed integer such that
, 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
.
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).