Strongly J-Clean Skew Triangular Matrix Rings

Let R be an arbitrary ring with identity. An element a ∈ R is strongly J-clean if there exist an idempotent e ∈ R and element w ∈ J(R) such that a = e + w and ew = ew. A ring R is strongly J-clean in case every element in R is strongly J-clean. In this note, we investigate the strong J-cleanness of the skew triangular matrix ring Tn(R, σ) over a local ring R, where σ is an endomorphism of R and n = 2, 3, 4.