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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].

MSC:

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