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.

Nahid Ashrafi, Marjan Sheibani, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195-363, Semnan 35131-19111, Iran, e-mail: n.ashrafi@semnan.ac.ir, m.sheibani1@gmail.com; Huanyin Chen (corresponding author), Department of Mathematics, Hangzhou Normal University, 16 Xuelin St, Jianggan, Hangzhou, 410006, Zhejiang, China, e-mail: huanyinchen@aliyun.com

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 2010): 16S34, 16U10, 16E50

DOI: 10.21136/CMJ.2017.0677-15

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