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Traces of multiadditive maps on rank-s matrices
Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2023-01-31 , DOI: 10.1080/03081087.2023.2172542
Haiyang Jiang 1 , Xiaowei Xu 1 , Haoran Yu 1
Affiliation  

Let m, n be integers such that 1<m<n. Let R=Mn(D) be the ring of all n×n matrices over a division ring D, M an additive subgroup of R and G:RmR an m-additive map. In this paper, under a mild technical assumption, we prove that δ1(x)=G(x,,x)M for each rank-s matrix xR implies δ1(x)M for each xR, where s is a fixed integer such that ms<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 ms<n.



中文翻译:

秩 s 矩阵上的多加性映射的迹

m , n为满足 1< m < n的整数。让R=n()成为所有人的戒指n×n除环上的矩阵,的加法子群RG:RR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δ1个(X)=G(X,……,X)对于每个等级矩阵XR暗示δ1个(X)每个XR,其中s是一个固定整数,使得<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<n.

更新日期:2023-01-31
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