## Strongly clean matrix rings over local rings.(English)Zbl 1117.16017

Summary: An element of a ring $$R$$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute, and $$R$$ is called strongly clean if every element of $$R$$ is strongly clean. In this paper, we determine when a $$2\times 2$$ matrix $$A$$ over a commutative local ring is strongly clean. Several equivalent criteria are given for such a matrix to be strongly clean. Consequently, we obtain several equivalent conditions for the $$2\times 2$$ matrix ring over a commutative local ring to be strongly clean, extending a result of J. Chen, X. Yang, and Y. Zhou [J. Algebra 301, No. 1, 280-293 (2006; Zbl 1110.16029)].

### MSC:

 16S50 Endomorphism rings; matrix rings 16U60 Units, groups of units (associative rings and algebras)

Zbl 1110.16029
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### References:

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