Strongly 2-nil-clean rings with involutions

Chen, Huanyin, and Sheibani Abdolyousefi, Marjan. "Strongly 2-nil-clean rings with involutions." Czechoslovak Mathematical Journal 69.2 (2019): 317-330. <http://eudml.org/doc/294440>.

@article{Chen2019,
abstract = {A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such rings and prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub ring and $J(R_2)$ is nil. The uniqueness of projections of such rings are thereby investigated.},
author = {Chen, Huanyin, Sheibani Abdolyousefi, Marjan},
journal = {Czechoslovak Mathematical Journal},
keywords = {nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring},
language = {eng},
number = {2},
pages = {317-330},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strongly 2-nil-clean rings with involutions},
url = {http://eudml.org/doc/294440},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Chen, Huanyin
AU - Sheibani Abdolyousefi, Marjan
TI - Strongly 2-nil-clean rings with involutions
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 317
EP - 330
AB - A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such rings and prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub ring and $J(R_2)$ is nil. The uniqueness of projections of such rings are thereby investigated.
LA - eng
KW - nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring
UR - http://eudml.org/doc/294440
ER -