Traces of multiadditive maps on rank-s matrices.

Let m, n be integers such that 1<m<n. Let $ \mathcal {R}=M_n(\mathbb {D}) $ R = M n (D) be the ring of all $ n\times n $ n × n matrices over a division ring $ \mathbb {D} $ D , $ \mathcal {M} $ M an additive subgroup of $ \mathcal {R} $ R and $ G:\mathcal {R}^m\rightarrow \mathcal {R} $ G : R m → R an m-additive map. In this paper, under a mild technical assumption, we prove that $ \delta _1(x)=G(x,\ldots,x)\in \mathcal {M} $ δ 1 (x) = G (x , ... , x) ∈ M for each rank-s matrix $ x\in \mathcal {R} $ x ∈ R implies $ \delta _1(x)\in \mathcal {M} $ δ 1 (x) ∈ M for each $ x\in \mathcal {R} $ x ∈ R , where s is a fixed integer such that $ m\leq s \lt n $ m ≤ 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\leq s \lt n $ m ≤ s < n. [ABSTRACT FROM AUTHOR]