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Certain additive decompositions in a noncommutative ring

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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 × 2 matrix A over a projective-free ring R is strongly J-clean if and only if AJ(M2(R)), or I2AJ(M2(R)), or A is similar to \(\left( {\matrix{ 0 & \lambda \cr 1 & \mu \cr } } \right)\), where λ ∈ J(R), μ ∈ 1 + J(R), and the equation x2 − λ = 0 has a root in J(R) and a root in 1 + J(R). We further prove that f(x) ∈ R[[x]] is strongly J-clean if f(0) ∈ R be optimally J-clean.

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The authors would like to thank the referee for his/her helpful comments leading to the improvement of this paper.

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Correspondence to Marjan Sheibani.

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This work was supported by the Natural Science Foundation of Zhejiang Province, China [grant number LY21A010018].

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Chen, H., Sheibani, M. & Bahmani, R. Certain additive decompositions in a noncommutative ring. Czech Math J 72, 1217–1226 (2022).

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