An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute . A ring is strongly J-clean in case each of its elements is strongly J-clean . A ring R has stable range one provided that aR + bR = R implies that there exists a y R such that a+by U(R). A long standing question asks whether strongly clean rings have stable range one . The other motivation of introducing strong J-cleanness is to construct a natural sub type="#_x0000_t75" for a large type="#_x0000_t75" denotes the ring of all upper triangular matrices over a ring R . Let R be a local ring , and let n ? 2 . It is proven that Tn(R) is strongly J-clean if and only if , R is bleached and R / J (R) . Further , we prove that the ring of all 2 × 2 matrices over a commutative local ring is not strongly J-clean . For local rings , we get criteria on the strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices . It is shown that the ring of 2 × 2 matrices over a commutative local ring is strongly clean if and only if every purely singular 2 × 2 matrix over them is strongly J-clean . The strong J-cleanness of a 2 × 2 matrix over a commutative local ring is completely characterized by means of a quadratic equation . In addition, it is shown that for a commutative local ring R, the following are equivalent: (1) A (R) is strongly J-clean; (2) A J ( (R)) or -A (R) or the equation has a root in J (R) and a root i 1+ J (R) .From these , we see that strong J-cleanness possesses many nice properties though , many of them could not be derived as done in other kind of cleanness .