Very cleanness of generalized matrices

An element a in a ring R is very clean in case there exists an idempotent e ∈ R such that ae = ea and either a − e or a + e is invertible. An element a in a ring R is very J -clean provided that there exists an idempotent e ∈ R such that ae = ea and either a − e ∈ J (R) or a + e ∈ J (R). Let R be a local ring, and let s ∈ C(R). We prove that A ∈ Ks(R) is very clean if and only if A ∈ U (Ks(R)), I ± A ∈ U (Ks(R)) or A ∈ Ks(R) is very J-clean.