Jianwei Zhu’s research while affiliated with Jilin University and other places

Let (Formula presented.) be integers, and let (Formula presented.) be the ring of all (Formula presented.) matrices over a field (Formula presented.) with centre (Formula presented.). Assume that (Formula presented.) is an m-additive map such that (Formula presented.) ((Formula presented.)), where (Formula presented.) denotes the trace map of G and (Formula presented.) is the set of all (Formula presented.) invertible matrices over (Formula presented.). Then (Formula presented.) ((Formula presented.)) if one of the followings holds: (1) char (Formula presented.); (2) char (Formula presented.); (3) char (Formula presented.) and (Formula presented.); (4) (Formula presented.). As applications, we give a slight improvement in the theorem by Franca and extend the condition from the prime ring (Formula presented.) to its nonadditive subset (Formula presented.) for the theorems by Lee et al. and Beidar et al.

Let n and s be integers such that $1\le s<\frac{n}{2}$, and let $M_n(\mathbb {K})$ be the ring of all $n\times n$ matrices over a field $\mathbb {K}$. Denote by $[\frac{n}{s}]$ the least integer m with $m\ge \frac{n}{s}$. In this short note, it is proved that if $g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})$ is a map such that $g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)$ holds for any $[\frac{n}{s}]$ rank-s matrices $A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})$, then g(x)=f(x)+g(0), $x\in M_n(\mathbb {K})$, for some additive map $f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})$. Particularly, g is additive if $char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right)$.