Geodesic
Certain additive decompositions in a noncommutative ring
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1217-1226
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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.
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.
DOI : 10.21136/CMJ.2022.0039-22
Classification : 15A09, 16E50, 16U60
Keywords: idempotent matrix; nilpotent matrix; projective-free ring; quadratic equation; power series 
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     title = {Certain additive decompositions in a noncommutative ring},
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     pages = {1217--1226},
     year = {2022},
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Chen, Huanyin; Sheibani, Marjan; Bahmani, Rahman. Certain additive decompositions in a noncommutative ring. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1217-1226. doi: 10.21136/CMJ.2022.0039-22

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