Certain additive decompositions in a noncommutative ring. (English) Zbl 07655796

Summary: 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 \(2\times 2\) 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 \(\left(\begin{smallmatrix}0&\lambda \\ 1&\mu\end{smallmatrix}\right)\), where \(\lambda\in J(R)\), \(\mu\in 1+J(R)\), 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.


15A09 Theory of matrix inversion and generalized inverses
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U60 Units, groups of units (associative rings and algebras)
Full Text: DOI


[1] Anderson, D. D.; Camillo, V. P., Commutative rings whose elements are a sum of a unit and idempotent, Commun. Algebra 30 (2002), 3327-3336 · Zbl 1083.13501 · doi:10.1081/AGB-120004490
[2] Ashrafi, N.; Nasibi, E., Strongly \(J\)-clean group rings, Proc. Rom. Acad., Ser. A, Math. Phys. Tech. Sci. Inf. Sci. 14 (2013), 9-12 · Zbl 1313.16036
[3] Chen, H., Rings Related Stable Range Conditions, Series in Algebra 11. World Scientific, Hackensack (2011) · Zbl 1245.16002 · doi:10.1142/8006
[4] Chen, H., Strongly \(J\)-clean matrices over local rings, Commun. Algebra 40 (2012), 1352-1362 · Zbl 1244.16024 · doi:10.1080/00927872.2010.551529
[5] Danchev, P. V.; McGovern, W. W., Commutative weakly nil clean unital rings, J. Algebra 425 (2015), 410-422 · Zbl 1316.16028 · doi:10.1016/j.jalgebra.2014.12.003
[6] Diesl, A. J.; Dorsey, T. J., Strongly clean matrices over arbitrary rings, J. Algebra 399 (2014), 854-869 · Zbl 1310.16023 · doi:10.1016/j.jalgebra.2013.08.044
[7] Dorsey, T. J., Cleanness and Strong Cleanness of Rings of Matrices: Ph.D. Thesis, University of California, Berkeley (2006)
[8] Fan, L.; Yang, X., A note on strongly clean matrix rings, Commun. Algebra 38 (2010), 799-806 · Zbl 1191.16029 · doi:10.1080/00927870802570693
[9] Koşan, M. T.; Yildirim, T.; Zhou, Y., Rings whose elements are the sum of a tripotent and an element from the Jacobson radical, Can. Math. Bull. 62 (2019), 810-821 · Zbl 1490.16091 · doi:10.4153/S0008439519000092
[10] Shifflet, D. R., Optimally Clean Rings: Ph.D. Thesis, Bowling Green State University, Bowling Green (2011)
[11] Yang, X.; Zhou, Y., Strong cleanness of the \(2\times 2\) matrix ring over a general local ring, J. Algebra 320 (2008), 2280-2290 · Zbl 1162.16017 · doi:10.1016/j.jalgebra.2008.06.012
[12] Zhu, H.; Zou, H.; Patrício, P., Generalized inverses and their relations with clean decompositions, J. Algebra Appl. 18 (2019), Article ID 1950133, 9 pages · Zbl 1453.16039 · doi:10.1142/S0219498819501330
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.