Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
idempotent element; nilpotent element; nil clean ring; weakly nil clean ring
idempotent element; nilpotent element; nil clean ring; weakly nil clean ring
Summary:
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 $n\geq 2$.
References:
[1] Ahn, M.-S., Anderson, D. D.: 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] Anderson, D. D., Camillo, V. P.: 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] Andrica, D., Călugăreanu, G.: 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] Breaz, S., Călugăreanu, G., Danchev, P., Micu, T.: Nil-clean matrix rings. Linear Algebra Appl. 439 (2013), 3115-3119. DOI 10.1016/j.laa.2013.08.027 | MR 3116417 | Zbl 06259710
[5] Breaz, S., Danchev, P., Zhou, Y.: 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] Burgess, W. D., Stephenson, W.: 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] Chacron, M.: 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] Chen, H.: Rings Related to Stable Range Conditions. Series in Algebra 11, World Scientific, Hackensack (2011). MR 2752904 | Zbl 1245.16002
[9] Danchev, P. V., McGovern, W. W.: 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] 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] Koşan, M. T., Lee, T.-K., Zhou, Y.: 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] McGovern, W. W., Raja, S., Sharp, A.: 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] Nicholson, W. K.: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229 (1977), 269-278. DOI 10.2307/1998510 | MR 0439876 | Zbl 0352.16006
[14] Yu, H.-P.: On quasi-duo rings. Glasg. Math. J. 37 (1995), 21-31. DOI 10.1017/S0017089500030342 | MR 1316960 | Zbl 0819.16001
Search
Partner of
