A subclass of strongly clean rings

Orhan Gurgun; Sait Halicioglu and Burcu Ungor

Communications in Mathematics (2015)

  • Volume: 23, Issue: 1, page 13-31
  • ISSN: 1804-1388

Abstract

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In this paper, we introduce a subclass of strongly clean rings. Let R be a ring with identity, J be the Jacobson radical of R , and let J # denote the set of all elements of R which are nilpotent in R / J . An element a ∈ R is called very J # -clean provided that there exists an idempotent e ∈ R such that a e = e a and a - e or a + e is an element of J # . A ring R is said to be very J # -clean in case every element in R is very J # -clean. We prove that every very J # -clean ring is strongly π -rad clean and has stable range one. It is shown that for a commutative local ring R , A ( x ) ∈ M 2 ( R [ [ x ] ] ) is very J # -clean if and only if A ( 0 ) ∈ M 2 ( R ) is very J # -clean. Various basic characterizations and properties of these rings are proved. We obtain a partial answer to the open question whether strongly clean rings have stable range one. This paper is dedicated to Professor Abdullah Harmanci on his 70th birthday

How to cite

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Gurgun, Orhan, and Ungor, Sait Halicioglu and Burcu. "A subclass of strongly clean rings." Communications in Mathematics 23.1 (2015): 13-31. <http://eudml.org/doc/271650>.

@article{Gurgun2015,
abstract = {In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^\{\#\}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very $J^\{\#\}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^\{\#\}$. A ring $R$ is said to be very $J^\{\#\}$-clean in case every element in $R$ is very $J^\{\#\}$-clean. We prove that every very $J^\{\#\}$-clean ring is strongly $\pi $-rad clean and has stable range one. It is shown that for a commutative local ring $R$, $A(x)\in M_2\big (R[[x]]\big )$ is very $J^\{\#\}$-clean if and only if $A(0)\in M_2(R)$ is very $J^\{\#\}$-clean. Various basic characterizations and properties of these rings are proved. We obtain a partial answer to the open question whether strongly clean rings have stable range one. This paper is dedicated to Professor Abdullah Harmanci on his 70th birthday},
author = {Gurgun, Orhan, Ungor, Sait Halicioglu and Burcu},
journal = {Communications in Mathematics},
keywords = {Very $J^\{\#\}$-clean matrix; very $J^\{\#\}$-clean ring; local ring; feckly reduced rings; Jacobson radical; -regular rings; exchange rings; semi-Abelian rings; quasi-duo rings},
language = {eng},
number = {1},
pages = {13-31},
publisher = {University of Ostrava},
title = {A subclass of strongly clean rings},
url = {http://eudml.org/doc/271650},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Gurgun, Orhan
AU - Ungor, Sait Halicioglu and Burcu
TI - A subclass of strongly clean rings
JO - Communications in Mathematics
PY - 2015
PB - University of Ostrava
VL - 23
IS - 1
SP - 13
EP - 31
AB - In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very $J^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be very $J^{\#}$-clean in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi $-rad clean and has stable range one. It is shown that for a commutative local ring $R$, $A(x)\in M_2\big (R[[x]]\big )$ is very $J^{\#}$-clean if and only if $A(0)\in M_2(R)$ is very $J^{\#}$-clean. Various basic characterizations and properties of these rings are proved. We obtain a partial answer to the open question whether strongly clean rings have stable range one. This paper is dedicated to Professor Abdullah Harmanci on his 70th birthday
LA - eng
KW - Very $J^{\#}$-clean matrix; very $J^{\#}$-clean ring; local ring; feckly reduced rings; Jacobson radical; -regular rings; exchange rings; semi-Abelian rings; quasi-duo rings
UR - http://eudml.org/doc/271650
ER -

References

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