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24 months)
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Keywords:
idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series
idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series
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 $\left (\smallmatrix 0&\lambda \\ 1&\mu \endsmallmatrix \right )$, 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.
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