Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1217-1226, 2022

Certain additive decompositions in a noncommutative ring

Huanyin Chen, Marjan Sheibani, Rahman Bahmani

Received January 30, 2022.   Published online July 21, 2022.

Abstract:  We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J (M_2(R))$, or $I_2-A\in J(M_2(R))$, or $A$ is similar to $\left(\smallmatrix0&\lambda
1&\mu\right)$, where $\lambda\in J(R)$, $\mu\in1+J(R)$, and the equation $x^2-x\mu-\lambda=0$ has a root in $J(R)$ and a root in $1+J(R)$. We further prove that $f(x)\in R[[x]]$ is strongly $J$-clean if $f(0)\in R$ be optimally $J$-clean.
Keywords:  idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series
Classification MSC:  15A09, 16E50, 16U60
DOI:  10.21136/CMJ.2022.0039-22

[1] 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
[2] N. Ashrafi, E. Nasibi: Strongly $J$-clean group rings. Proc. Rom. Acad., Ser. A, Math. Phys. Tech. Sci. Inf. Sci. 14 (2013), 9-12. MR 3038863 | Zbl 1313.16036
[3] H. Chen: Rings Related Stable Range Conditions. Series in Algebra 11. World Scientific, Hackensack (2011). DOI 10.1142/8006 | MR 2752904 | Zbl 1245.16002
[4] H. Chen: Strongly $J$-clean matrices over local rings. Commun. Algebra 40 (2012), 1352-1362. DOI 10.1080/00927872.2010.551529 | MR 2912989 | Zbl 1244.16024
[5] P. V. Danchev, W. W. McGovern: Commutative weakly nil clean unital rings. J. Algebra 425 (2015), 410-422. DOI 10.1016/j.jalgebra.2014.12.003 | MR 3295991 | Zbl 1316.16028
[6] A. J. Diesl, T. J. Dorsey: Strongly clean matrices over arbitrary rings. J. Algebra 399 (2014), 854-869. DOI 10.1016/j.jalgebra.2013.08.044 | MR 3144615 | Zbl 1310.16023
[7] T. J. Dorsey: Cleanness and Strong Cleanness of Rings of Matrices: Ph.D. Thesis. University of California, Berkeley (2006). MR 2709133
[8] L. Fan, X. Yang: A note on strongly clean matrix rings. Commun. Algebra 38 (2010), 799-806. DOI 10.1080/00927870802570693 | MR 2650370 | Zbl 1191.16029
[9] M. T. Koşan, T. Yildirim, Y. Zhou: Rings whose elements are the sum of a tripotent and an element from the Jacobson radical. Can. Math. Bull. 62 (2019), 810-821. DOI 10.4153/S0008439519000092 | MR 4028489 | Zbl 07128566
[10] D. R. Shifflet: Optimally Clean Rings: Ph.D. Thesis. Bowling Green State University, Bowling Green (2011). MR 3121884
[12] X. Yang, Y. Zhou: Strong cleanness of the $2\times 2$ matrix ring over a general local ring. J. Algebra 320 (2008), 2280-2290. DOI 10.1016/j.jalgebra.2008.06.012 | MR 2437500 | Zbl 1162.16017
[13] H. Zhu, H. Zou, P. Patricio: Generalized inverses and their relations with clean decompositions. J. Algebra Appl. 18 (2019), Article ID 1950133, 9 pages. DOI 10.1142/S0219498819501330 | MR 3977794 | Zbl 1453.16039

Affiliations:   Huanyin Chen, School of Mathematics, Hangzhou Normal University, No. 2318, Yuhangtang Road, 311121, Hangzhou, P. R. China, e-mail:; Marjan Sheibani (corresponding author), Farzanegan Campus, Semnan University, 17 Shahrivar Blvd. Semnan, Iran, e-mail:; Rahman Bahmani, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195-363, Semnan, Iran, e-mail:

PDF available at: