## 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.

### MSC:

 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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### References:

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