Geodesic
Certain decompositions of matrices over Abelian rings
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 417-425
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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 $n\geq 2$.
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 $n\geq 2$.
DOI : 10.21136/CMJ.2017.0677-15
Classification : 16E50, 16S34, 16U10
Keywords: idempotent element; nilpotent element; nil clean ring; weakly nil clean ring 
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Ashrafi, Nahid; Sheibani, Marjan; Chen, Huanyin. Certain decompositions of matrices over Abelian rings. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 417-425. doi: 10.21136/CMJ.2017.0677-15

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