Geodesic
A subclass of strongly clean rings
Communications in Mathematics, Tome 23 (2015) no. 1, pp. 13-31 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 \emph {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 \emph {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
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 \emph {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 \emph {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
Classification : 15A13, 15B99, 16L99
Keywords: Very $J^{\#}$-clean matrix; very $J^{\#}$-clean ring; local ring. 
@article{COMIM_2015_23_1_a1,
     author = {Gurgun, Orhan and Ungor, Sait Halicioglu and Burcu},
     title = {A subclass of strongly clean rings},
     journal = {Communications in Mathematics},
     pages = {13--31},
     year = {2015},
     volume = {23},
     number = {1},
     mrnumber = {3394075},
     zbl = {1347.16038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2015_23_1_a1/}
}
TY  - JOUR
AU  - Gurgun, Orhan
AU  - Ungor, Sait Halicioglu and Burcu
TI  - A subclass of strongly clean rings
JO  - Communications in Mathematics
PY  - 2015
SP  - 13
EP  - 31
VL  - 23
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/COMIM_2015_23_1_a1/
LA  - en
ID  - COMIM_2015_23_1_a1
ER  - 
%0 Journal Article
%A Gurgun, Orhan
%A Ungor, Sait Halicioglu and Burcu
%T A subclass of strongly clean rings
%J Communications in Mathematics
%D 2015
%P 13-31
%V 23
%N 1
%U http://geodesic.mathdoc.fr/item/COMIM_2015_23_1_a1/
%G en
%F COMIM_2015_23_1_a1
Gurgun, Orhan; Ungor, Sait Halicioglu and Burcu. A subclass of strongly clean rings. Communications in Mathematics, Tome 23 (2015) no. 1, pp. 13-31. http://geodesic.mathdoc.fr/item/COMIM_2015_23_1_a1/

[1] Agayev, N., Harmanci, A., Halicioglu, S.: On abelian rings. Turk J. Math., 34, 2010, 465-474, | MR | Zbl

[2] Anderson, D. D., Camillo, V. P.: Commutative rings whose elements are a sum of a unit and idempotent. Comm. Algebra, 30, 7, 2002, 3327-3336, | DOI | MR | Zbl

[3] Ara, P.: Strongly $\pi $-regular rings have stable range one. Proc. Amer. Math. Soc., 124, 1996, 3293-3298, | DOI | MR | Zbl

[4] Borooah, G., Diesl, A. J., Dorsey, T. J.: Strongly clean matrix rings over commutative local rings. J. Pure Appl. Algebra, 212, 1, 2008, 281-296, | MR | Zbl

[5] Chen, H.: On strongly $J$-clean rings. Comm. Algebra, 38, 2010, 3790-3804, | DOI | MR | Zbl

[6] Chen, H.: Rings related to stable range conditions. 11, 2011, World Scientific, Hackensack, NJ, | MR | Zbl

[7] Chen, H., Kose, H., Kurtulmaz, Y.: Factorizations of matrices over projective-free rings. arXiv preprint arXiv:1406.1237, 2014, | MR

[8] Chen, H., Ungor, B., Halicioglu, S.: Very clean matrices over local rings. arXiv preprint arXiv:1406.1240, 2014,

[9] Diesl, A. J.: Classes of strongly clean rings. 2006, ProQuest, Ph.D. Thesis, University of California, Berkeley.. | MR

[10] Diesl, A. J.: Nil clean rings. J. Algebra, 383, 2013, 197-211, | DOI | MR | Zbl

[11] Evans, E. G.: Krull-Schmidt and cancellation over local rings. Pacific J. Math., 46, 1973, 115-121, | DOI | MR | Zbl

[12] Han, J., Nicholson, W. K.: Extensions of clean rings. Comm. Algebra, 29, 2011, 2589-2595, | DOI | MR

[13] Herstein, I. N.: Noncommutative rings, The Carus Mathematical Monographs. 15, 1968, Published by The Mathematical Association of America, Distributed by John Wiley and Sons, Inc., New York, 1968.. | MR | Zbl

[14] Lam, T. Y.: A first course in noncommutative rings. 131, 2001, Graduate Texts in Mathematics, Springer-Verlag, New York, | MR | Zbl

[15] Mesyan, Z.: The ideals of an ideal extension. J. Algebra Appl., 9, 2010, 407-431, | DOI | MR | Zbl

[16] Nicholson, W. K.: Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 229, 1977, 269-278, | DOI | MR | Zbl

[17] Nicholson, W. K.: Strongly clean rings and Fitting's lemma. Comm. Algebra, 27, 1999, 3583-3592, | DOI | MR | Zbl

[18] Nicholson, W. K., Zhou, Y.: Rings in which elements are uniquely the sum of an idempotent and a unit. Glasgow Math. J., 46, 2004, 227-236, | DOI | MR | Zbl

[19] Vaserstein, L. N.: Bass's first stable range condition. J. Pure Appl. Algebra, 34, 2, 1984, 319-330, | MR | Zbl