Certain decompositions of matrices over Abelian rings

Czechoslovak Mathematical Journal

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen¹ 

关键词: idempotent element;    nilpotent element;    nil clean ring;    weakly nil clean ring;   

DOI  :  10.21136/CMJ.2017.0677-15

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n\in{\Bbb N}$. We prove that $M_n(R)$ is nil clean if and only if $R/J(R)$ is Boolean and $M_n(J(R))$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J(R)$ is ${\Bbb Z}_3$, $B$ or ${\Bbb Z}_3\oplus B$ where $B$ is a Boolean ring, and that $M_n(R)$ is weakly nil clean if and only if $M_n(R)$ is nil clean for all $n\geq2$.

【 授权许可】

Unknown