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Keywords:
Very $J^{\#}$-clean matrix; very $J^{\#}$-clean ring; local ring.
Very $J^{\#}$-clean matrix; very $J^{\#}$-clean ring; local ring.
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
References:
[1] Agayev, N., Harmanci, A., Halicioglu, S.: On abelian rings. Turk J. Math., 34, 2010, 465-474, MR 2721960 | Zbl 1210.16037
[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 10.1081/AGB-120004490 | MR 1914999 | Zbl 1083.13501
[3] Ara, P.: Strongly $\pi $-regular rings have stable range one. Proc. Amer. Math. Soc., 124, 1996, 3293-3298, DOI 10.1090/S0002-9939-96-03473-9 | MR 1343679 | Zbl 0865.16007
[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 2355051 | Zbl 1162.16016
[5] Chen, H.: On strongly $J$-clean rings. Comm. Algebra, 38, 2010, 3790-3804, DOI 10.1080/00927870903286835 | MR 2760691 | Zbl 1242.16026
[6] Chen, H.: Rings related to stable range conditions. 11, 2011, World Scientific, Hackensack, NJ, MR 2752904 | Zbl 1245.16002
[7] Chen, H., Kose, H., Kurtulmaz, Y.: Factorizations of matrices over projective-free rings. arXiv preprint arXiv:1406.1237, 2014, MR 3439874
[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 2709132
[10] Diesl, A. J.: Nil clean rings. J. Algebra, 383, 2013, 197-211, DOI 10.1016/j.jalgebra.2013.02.020 | MR 3037975 | Zbl 1296.16016
[11] Evans, E. G.: Krull-Schmidt and cancellation over local rings. Pacific J. Math., 46, 1973, 115-121, DOI 10.2140/pjm.1973.46.115 | MR 0323815 | Zbl 0272.13006
[12] Han, J., Nicholson, W. K.: Extensions of clean rings. Comm. Algebra, 29, 2011, 2589-2595, DOI 10.1081/AGB-100002409 | MR 1845131
[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 1449137 | Zbl 0177.05801
[14] Lam, T. Y.: A first course in noncommutative rings. 131, 2001, Graduate Texts in Mathematics, Springer-Verlag, New York, MR 1838439 | Zbl 0980.16001
[15] Mesyan, Z.: The ideals of an ideal extension. J. Algebra Appl., 9, 2010, 407-431, DOI 10.1142/S0219498810003999 | MR 2659728 | Zbl 1200.16042
[16] Nicholson, W. K.: Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 229, 1977, 269-278, DOI 10.1090/S0002-9947-1977-0439876-2 | MR 0439876 | Zbl 0352.16006
[17] Nicholson, W. K.: Strongly clean rings and Fitting's lemma. Comm. Algebra, 27, 1999, 3583-3592, DOI 10.1080/00927879908826649 | MR 1699586 | Zbl 0946.16007
[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 10.1017/S0017089504001727 | MR 2062606 | Zbl 1057.16007
[19] Vaserstein, L. N.: Bass's first stable range condition. J. Pure Appl. Algebra, 34, 2, 1984, 319-330, MR 0772066 | Zbl 0547.16017
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