Traces of multiadditive maps on rank-s matrices

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 .