Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 417-425, 2017

Certain decompositions of matrices over Abelian rings

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen

Received December 14, 2015.  First published March 1, 2017.

Abstract:  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 $n\geq2$.
Keywords:  idempotent element; nilpotent element; nil clean ring; weakly nil clean ring
Classification MSC:  16S34, 16U10, 16E50
DOI:  10.21136/CMJ.2017.0677-15

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Affiliations:   Nahid Ashrafi, Marjan Sheibani, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195-363, Semnan 35131-19111, Iran, e-mail:,; Huanyin Chen (corresponding author), Department of Mathematics, Hangzhou Normal University, 16 Xuelin St, Jianggan, Hangzhou, 410006, Zhejiang, China, e-mail:

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