Strongly 2-nil-clean rings with involutions
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 317-330
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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.
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.
DOI :
10.21136/CMJ.2018.0291-17
Classification :
16E50, 16U99, 16W10
Keywords: nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring
Keywords: nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring
@article{10_21136_CMJ_2018_0291_17,
author = {Chen, Huanyin and Sheibani Abdolyousefi, Marjan},
title = {Strongly 2-nil-clean rings with involutions},
journal = {Czechoslovak Mathematical Journal},
pages = {317--330},
year = {2019},
volume = {69},
number = {2},
doi = {10.21136/CMJ.2018.0291-17},
mrnumber = {3959946},
zbl = {07088786},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0291-17/}
}
TY - JOUR AU - Chen, Huanyin AU - Sheibani Abdolyousefi, Marjan TI - Strongly 2-nil-clean rings with involutions JO - Czechoslovak Mathematical Journal PY - 2019 SP - 317 EP - 330 VL - 69 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0291-17/ DO - 10.21136/CMJ.2018.0291-17 LA - en ID - 10_21136_CMJ_2018_0291_17 ER -
%0 Journal Article %A Chen, Huanyin %A Sheibani Abdolyousefi, Marjan %T Strongly 2-nil-clean rings with involutions %J Czechoslovak Mathematical Journal %D 2019 %P 317-330 %V 69 %N 2 %U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0291-17/ %R 10.21136/CMJ.2018.0291-17 %G en %F 10_21136_CMJ_2018_0291_17
Chen, Huanyin; Sheibani Abdolyousefi, Marjan. Strongly 2-nil-clean rings with involutions. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 317-330. doi: 10.21136/CMJ.2018.0291-17
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