Certain additive decompositions in a noncommutative ring

Abstract

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 A ∈ J(M₂(R)), or I₂ − A ∈ J(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 − xμ − 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.