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 $m\le s<n$.

令m , n为满足 1< m < n的整数。让$\mathcal{R}={米}_n(\mathbb{丁})$成为所有人的戒指$n\times n$除环上的矩阵$\mathbb{丁}$,$\mathcal{米}$的加法子群$\mathcal{R}$和$G:{\mathcal{R}}^{米}\to \mathcal{R}$一个m加法映射。在本文中，在温和的技术假设下，我们证明了${\delta }_{\mathrm{1个}}(X)=G(X,\dots \dots ,X)\in \mathcal{米}$对于每个等级矩阵$X\in \mathcal{R}$暗示${\delta }_{\mathrm{1个}}(X)\in \mathcal{米}$每个$X\in \mathcal{R}$，其中s是一个固定整数，使得$米\le 秒<n$，在 [Xu X, Zhu J., Central traces of multiadditive maps on invertible matrices, Linear Multilinear Algebra 2018 中考虑了s  =  n的情况；66:1442–1448]。此外，还提供了一个示例，表明如果s < m则结论不成立。作为应用，我们还扩展了 Liu、Franca 等人、Lee 等人的结论。和 Beidar 等人，分别针对秩矩阵的情况$米\le 秒<n$.