| Title:
|
A subclass of strongly clean rings (English) |
| Author:
|
Gurgun, Orhan |
| Author:
|
Ungor, Sait Halicioglu and Burcu |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
23 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
13-31 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 (English) |
| Keyword:
|
Very $J^{\#}$-clean matrix |
| Keyword:
|
very $J^{\#}$-clean ring |
| Keyword:
|
local ring. |
| MSC:
|
15A13 |
| MSC:
|
15B99 |
| MSC:
|
16L99 |
| idZBL:
|
Zbl 1347.16038 |
| idMR:
|
MR3394075 |
| . |
| Date available:
|
2015-08-25T13:56:10Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144356 |
| . |
| Reference:
|
[1] Agayev, N., Harmanci, A., Halicioglu, S.: On abelian rings.Turk J. Math., 34, 2010, 465-474, Zbl 1210.16037, MR 2721960 |
| Reference:
|
[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, Zbl 1083.13501, MR 1914999, 10.1081/AGB-120004490 |
| Reference:
|
[3] Ara, P.: Strongly $\pi $-regular rings have stable range one.Proc. Amer. Math. Soc., 124, 1996, 3293-3298, Zbl 0865.16007, MR 1343679, 10.1090/S0002-9939-96-03473-9 |
| Reference:
|
[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, Zbl 1162.16016, MR 2355051 |
| Reference:
|
[5] Chen, H.: On strongly $J$-clean rings.Comm. Algebra, 38, 2010, 3790-3804, Zbl 1242.16026, MR 2760691, 10.1080/00927870903286835 |
| Reference:
|
[6] Chen, H.: Rings related to stable range conditions.11, 2011, World Scientific, Hackensack, NJ, Zbl 1245.16002, MR 2752904 |
| Reference:
|
[7] Chen, H., Kose, H., Kurtulmaz, Y.: Factorizations of matrices over projective-free rings.arXiv preprint arXiv:1406.1237, 2014, MR 3439874 |
| Reference:
|
[8] Chen, H., Ungor, B., Halicioglu, S.: Very clean matrices over local rings.arXiv preprint arXiv:1406.1240, 2014, |
| Reference:
|
[9] Diesl, A. J.: Classes of strongly clean rings.2006, ProQuest, Ph.D. Thesis, University of California, Berkeley.. MR 2709132 |
| Reference:
|
[10] Diesl, A. J.: Nil clean rings.J. Algebra, 383, 2013, 197-211, Zbl 1296.16016, MR 3037975, 10.1016/j.jalgebra.2013.02.020 |
| Reference:
|
[11] Evans, E. G.: Krull-Schmidt and cancellation over local rings.Pacific J. Math., 46, 1973, 115-121, Zbl 0272.13006, MR 0323815, 10.2140/pjm.1973.46.115 |
| Reference:
|
[12] Han, J., Nicholson, W. K.: Extensions of clean rings.Comm. Algebra, 29, 2011, 2589-2595, MR 1845131, 10.1081/AGB-100002409 |
| Reference:
|
[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.. Zbl 0177.05801, MR 1449137 |
| Reference:
|
[14] Lam, T. Y.: A first course in noncommutative rings.131, 2001, Graduate Texts in Mathematics, Springer-Verlag, New York, Zbl 0980.16001, MR 1838439 |
| Reference:
|
[15] Mesyan, Z.: The ideals of an ideal extension.J. Algebra Appl., 9, 2010, 407-431, Zbl 1200.16042, MR 2659728, 10.1142/S0219498810003999 |
| Reference:
|
[16] Nicholson, W. K.: Lifting idempotents and exchange rings.Trans. Amer. Math. Soc., 229, 1977, 269-278, Zbl 0352.16006, MR 0439876, 10.1090/S0002-9947-1977-0439876-2 |
| Reference:
|
[17] Nicholson, W. K.: Strongly clean rings and Fitting's lemma.Comm. Algebra, 27, 1999, 3583-3592, Zbl 0946.16007, MR 1699586, 10.1080/00927879908826649 |
| Reference:
|
[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, Zbl 1057.16007, MR 2062606, 10.1017/S0017089504001727 |
| Reference:
|
[19] Vaserstein, L. N.: Bass's first stable range condition.J. Pure Appl. Algebra, 34, 2, 1984, 319-330, Zbl 0547.16017, MR 0772066 |
| . |
