## When is the $$2\times 2$$ matrix ring over a commutative local ring strongly clean?(English)Zbl 1110.16029

Throughout all rings are associative with identity. A ring $$R$$ is said to be strongly clean if every element of $$R$$ is the sum of an idempotent and a unit that commute. Local rings are obviously strongly clean. Let $$U(R)$$ be the group of units of a ring $$R$$, $$J(R)$$ be the Jacobson radical of $$R$$ and $$C_2$$ be the cyclic group of order 2.
The main results established by the authors are the following. (1) Let $$R$$ be a commutative ring. If $$M_2(R)$$ is strongly clean, then for all $$w\in J(R)$$ the equation $$x^2-x=w$$ is solvable in $$R$$. (2) Let $$R$$ be a commutative local ring and let $$n\geq 1$$. Then: (i) if either $$2\in U(R)$$ or $$R/J(R)\cong\mathbb{Z}_2$$, then $$M_2(R)$$ is strongly clean if and only if for all $$w\in J(R)$$ the equation $$x^2-x=w$$ is solvable in $$R$$; (ii) if $$2\in J(R)$$ and $$R/J(R)\not\cong\mathbb{Z}_2$$, then $$M_2(R)$$ is strongly clean if and only if for all $$w_1,w_2\in J(R)$$ the equation $$x^2+(1+w_1)x=w_2$$ is solvable in $$R$$; (iii) $$M_2(R)$$ is strongly clean if and only if $$M_2(R[x]/(x^n))$$ is strongly clean if and only if $$M_2(RC_2)$$ is strongly clean.

### MSC:

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

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