Strong cleanness of the \(2\times 2\) matrix ring over a general local ring. (English) Zbl 1162.16017

Throughout \(R\) is an associative ring with identity. A ring is called strongly clean if every element is the sum of an idempotent and a unit that commute with each other.
The authors deal with the problem of completely characterizing the local rings for which the matrix ring \(\mathbb{M}_n(R)\) is strongly clean, and solve the case \(n=2\). They also give applications and examples, and give a characterization when the \(2\times 2\) matrix ring over a local ring is strongly \(\pi\)-regular.


16S50 Endomorphism rings; matrix rings
16U60 Units, groups of units (associative rings and algebras)
16L30 Noncommutative local and semilocal rings, perfect rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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