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Certain additive decompositions in a noncommutative ring. (English) Zbl 07655796

Summary: We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a \(2\times 2\) matrix \(A\) over a projective-free ring \(R\) is strongly \(J\)-clean if and only if \(A\in J(M_2(R))\), or \(I_2-A\in J(M_2(R))\), or \(A\) is similar to \(\left(\begin{smallmatrix}0&\lambda \\ 1&\mu\end{smallmatrix}\right)\), where \(\lambda\in J(R)\), \(\mu\in 1+J(R)\), and the equation \(x^2-x\mu -\lambda =0\) has a root in \(J(R)\) and a root in \(1+J(R)\). We further prove that \(f(x)\in R[[x]]\) is strongly \(J\)-clean if \(f(0)\in R\) be optimally \(J\)-clean.

MSC:

15A09 Theory of matrix inversion and generalized inverses
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U60 Units, groups of units (associative rings and algebras)
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