## Strongly clean matrix rings over commutative local rings.(English)Zbl 1162.16016

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 completely characterize the commutative local rings $$R$$ for which the matrix ring $$\mathbb{M}_n(R)$$ is strongly clean, in terms of factorization in $$R[t]$$. Among several other results, it is shown that for any monic polynomial $$f\in R[t]$$ , the strong cleanness of the companion matrix of $$f$$ is equivalent to the strong cleanness of all matrices with characteristic polynomial $$f$$. Various interesting examples are given and open questions are raised.

### MSC:

 16S50 Endomorphism rings; matrix rings 16U60 Units, groups of units (associative rings and algebras) 13H99 Local rings and semilocal rings
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### References:

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