Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 417-425, 2017

Certain decompositions of matrices over Abelian rings

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen

Received December 14, 2015.   First published March 1, 2017.

Nahid Ashrafi, Marjan Sheibani, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195-363, Semnan 35131-19111, Iran, e-mail: n.ashrafi@semnan.ac.ir, m.sheibani1@gmail.com; Huanyin Chen (corresponding author), Department of Mathematics, Hangzhou Normal University, 16 Xuelin St, Jianggan, Hangzhou, 410006, Zhejiang, China, e-mail: huanyinchen@aliyun.com

Abstract: A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n\in{\Bbb N}$. We prove that $M_n(R)$ is nil clean if and only if $R/J(R)$ is Boolean and $M_n(J(R))$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J(R)$ is ${\Bbb Z}_3$, $B$ or ${\Bbb Z}_3\oplus B$ where $B$ is a Boolean ring, and that $M_n(R)$ is weakly nil clean if and only if $M_n(R)$ is nil clean for all $n\geq2$.

Keywords: idempotent element; nilpotent element; nil clean ring; weakly nil clean ring

Classification (MSC 2010): 16S34, 16U10, 16E50

DOI: 10.21136/CMJ.2017.0677-15

Full text available as PDF.


References:
  [1] M.-S. Ahn, D. D. Anderson: Weakly clean rings and almost clean rings. Rocky Mt. J. Math. 36 (2006), 783-798. DOI 10.1216/rmjm/1181069429 | MR 2254362 | Zbl 1131.13301
  [2] D. D. Anderson, V. P. Camillo: Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30 (2002), 3327-3336. DOI 10.1081/AGB-120004490 | MR 1914999 | Zbl 1083.13501
  [3] D. Andrica, G. Călugăreanu: A nil-clean $2\times 2$ matrix over the integers which is not clean. J. Algebra Appl. 13 (2014), Article ID 1450009, 9 pages. DOI 10.1142/S0219498814500091 | MR 3195166 | Zbl 1294.16019
  [4] S. Breaz, G. Călugăreanu, P. Danchev, T. Micu: Nil-clean matrix rings. Linear Algebra Appl. 439 (2013), 3115-3119. DOI 10.1016/j.laa.2013.08.027 | MR 3116417 | Zbl 06259710
  [5] S. Breaz, P. Danchev, Y. Zhou: Rings in which every element is either a sum or a difference of a nilpotent and an idempotent. J. Algebra Appl. 15 (2016), Article ID 1650148, 11 pages. DOI 10.1142/S0219498816501486 | MR 3528770 | Zbl 06619808
  [6] W. D. Burgess, W. Stephenson: Rings all of whose Pierce stalks are local. Canad. Math. Bull. 22 (1979), 159-164. DOI 10.4153/CMB-1979-022-8 | MR 0537296 | Zbl 0411.16009
  [7] M. Chacron: On a theorem of Herstein. Can. J. Math. 21 (1969), 1348-1353. DOI 10.4153/CJM-1969-148-5 | MR 0262295 | Zbl 0213.04302
  [8] H. Chen: Rings Related to Stable Range Conditions. Series in Algebra 11, World Scientific, Hackensack (2011). MR 2752904 | Zbl 1245.16002
  [9] P. V. Danchev, W. W. McGovern: Commutative weakly nil clean unital rings. J. Algebra Appl. 425 (2015), 410-422. DOI 10.1016/j.jalgebra.2014.12.003 | MR 3295991 | Zbl 1316.16028
  [10] A. J. Diesl: Nil clean rings. J. Algebra 383 (2013), 197-211. DOI 10.1016/j.jalgebra.2013.02.020 | MR 3037975 | Zbl 1296.16016
  [11] M. T. Koşan, T.-K. Lee, Y. Zhou: When is every matrix over a division ring a sum of an idempotent and a nilpotent?. Linear Algebra Appl. 450 (2014), 7-12. DOI 10.1016/j.laa.2014.02.047 | MR 3192466 | Zbl 1303.15016
  [12] W. W. McGovern, S. Raja, A. Sharp: Commutative nil clean group rings. J. Algebra Appl. 14 (2015), Article ID 1550094, 5 pages. DOI 10.1142/S0219498815500942 | MR 3338090 | Zbl 1325.16024
  [13] W. K. Nicholson: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229 (1977), 269-278. DOI 10.2307/1998510 | MR 0439876 | Zbl 0352.16006
  [14] H.-P. Yu: On quasi-duo rings. Glasg. Math. J. 37 (1995), 21-31. DOI 10.1017/S0017089500030342 | MR 1316960 | Zbl 0819.16001


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