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.


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