Certain additive decompositions in a noncommutative ring

@article{Chen2022,
abstract = {We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J (M_2(R))$, or $I_2-A\in J(M_2(R))$, or $A$ is similar to $\left(\{\textstyle \begin\{matrix\}0&\lambda \\ 1&\mu \end\{matrix\}\}\right)$, where $\lambda \in J(R)$, $\mu \in 1+J(R)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J(R)$ and a root in $1+J(R)$. We further prove that $f(x)\in R[[x]]$ is strongly $J$-clean if $f(0)\in R$ be optimally $J$-clean.},
author = {Chen, Huanyin, Sheibani, Marjan, Bahmani, Rahman},
journal = {Czechoslovak Mathematical Journal},
keywords = {idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series},
language = {eng},
number = {4},
pages = {1217-1226},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Certain additive decompositions in a noncommutative ring},
url = {http://eudml.org/doc/298888},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Chen, Huanyin
AU - Sheibani, Marjan
AU - Bahmani, Rahman
TI - Certain additive decompositions in a noncommutative ring
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1217
EP - 1226
AB - We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J (M_2(R))$, or $I_2-A\in J(M_2(R))$, or $A$ is similar to $\left({\textstyle \begin{matrix}0&\lambda \\ 1&\mu \end{matrix}}\right)$, where $\lambda \in J(R)$, $\mu \in 1+J(R)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J(R)$ and a root in $1+J(R)$. We further prove that $f(x)\in R[[x]]$ is strongly $J$-clean if $f(0)\in R$ be optimally $J$-clean.
LA - eng
KW - idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series
UR - http://eudml.org/doc/298888
ER -