Certain decompositions of matrices over Abelian rings

Ashrafi, Nahid, Sheibani, Marjan, and Chen, Huanyin. "Certain decompositions of matrices over Abelian rings." Czechoslovak Mathematical Journal 67.2 (2017): 417-425. <http://eudml.org/doc/288185>.

@article{Ashrafi2017,
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 $n\in \{\mathbb \{N\}\}$. 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 $\{\mathbb \{Z\}\}_3$, $B$ or $\{\mathbb \{Z\}\}_3\oplus B$ 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 $n\ge 2$.},
author = {Ashrafi, Nahid, Sheibani, Marjan, Chen, Huanyin},
journal = {Czechoslovak Mathematical Journal},
keywords = {idempotent element; nilpotent element; nil clean ring; weakly nil clean ring},
language = {eng},
number = {2},
pages = {417-425},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Certain decompositions of matrices over Abelian rings},
url = {http://eudml.org/doc/288185},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Ashrafi, Nahid
AU - Sheibani, Marjan
AU - Chen, Huanyin
TI - Certain decompositions of matrices over Abelian rings
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 417
EP - 425
AB - 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 $n\in {\mathbb {N}}$. 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 ${\mathbb {Z}}_3$, $B$ or ${\mathbb {Z}}_3\oplus B$ 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 $n\ge 2$.
LA - eng
KW - idempotent element; nilpotent element; nil clean ring; weakly nil clean ring
UR - http://eudml.org/doc/288185
ER -