Strongly \(J\)-clean matrices over local rings. (English) Zbl 1244.16024

Summary: An element of a ring is called strongly \(J\)-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly \(J\)-clean \(2\times 2\) matrix over a noncommutative local ring. Criteria on strong \(J\)-cleanness of \(2\times 2\) matrices in terms of a quadratic equation are given. These extend the corresponding results of B. Li [Bull. Korean Math. Soc. 46, No. 1, 71-78 (2009; Zbl 1168.16014), Theorems 2.7 and 3.2], Y. Li [J. Algebra 312, No. 1, 397-404 (2007; Zbl 1117.16017), Theorem 2.6], and X. Yang and Y. Zhou [J. Algebra 320, No. 6, 2280-2290 (2008; Zbl 1162.16017), Theorem 7].


16S50 Endomorphism rings; matrix rings
16U60 Units, groups of units (associative rings and algebras)
16L30 Noncommutative local and semilocal rings, perfect rings
Full Text: DOI


[1] DOI: 10.1016/j.jalgebra.2006.10.029 · Zbl 1144.16023
[2] DOI: 10.1016/j.jpaa.2007.05.020 · Zbl 1162.16016
[3] DOI: 10.1016/j.jalgebra.2005.08.005 · Zbl 1110.16029
[4] DOI: 10.1080/00927870903286835 · Zbl 1242.16026
[5] DOI: 10.1142/9789814329729
[6] Diesl A. J., Classes of Strongly Clean Rings (2006)
[7] DOI: 10.1017/S0017089506003284 · Zbl 1114.16025
[8] DOI: 10.4134/BKMS.2009.46.1.071 · Zbl 1168.16014
[9] DOI: 10.1016/j.jalgebra.2006.10.032 · Zbl 1117.16017
[10] DOI: 10.1080/00927879908826649 · Zbl 0946.16007
[11] DOI: 10.1016/j.jalgebra.2008.06.012 · Zbl 1162.16017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.