arXiv:1409.3974v2 [math.RA] 28 Oct 2014 Strongly J-clean matrices over 2-projective-free rings Marjan Sheibani, Huanyin Chen and Rahman Bahmani Abstract An element a of a ring is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean 2 × 2 matrices over noncommutative 2-projective-free rings. For a 2-projective-free ring R, A ∈ M 2 (R) is strongly J-clean if and only if A ∈ J ( M 2 (R) ) , or I 2 − A ∈ J ( M 2 (R) ) , or A is similar to  0 λ 1 µ  where λ ∈ J (R),µ ∈ 1+ J (R), and the equation x 2 − xµ − λ = 0 has a root in J (R) and a root in 1 + J (R). Strongly J-clean 2 × 2 matrices over power series are therefrom investigated. We prove that if R is a 2-projective-free wb-ring then A(x) ∈ M 2 (R[[x]]) is strongly J-clean if and only if A(0) ∈ M 2 (R) is strongly J-clean. Keywords: strongly J-clean matrix; 2-projective-free ring; quadratic equation, power series. 2010 Mathematics Subject Classification: 15E50, 16U60. 1 Introduction An element a ∈ R is called strongly J-clean (clean) if there exists an idempotent e ∈ R such that a − e ∈ J (R)(U (R)) and ae = ea. A ring R is strongly J-clean (clean) provided that every element in R is strongly J-clean (clean). A ring R in uniquely clean if for any a ∈ R there exists a unique idempotent e ∈ R such that a − e ∈ U (R), which was introduced by Anderson and Camillo [1]. Nicholson and Zhou proved that a ring R is uniquely clean if and only if for any a ∈ R there exists a unique idempotent e ∈ R such that a − e ∈ J (R) [10, Theorem 20]. Evidently, { uniquely clean rings }  { strongly J-clean rings }  { strongly clean rings }. The inclusions are both proper. For instances, The ring T 2 (Z 2 ) of all 2 × 2 up triangular matrices over Z 2 is strongly J-clean, while it is not uniquely clean [4, Corollary 16.4.24]; and that Z 3 is strongly clean, while it is not strongly J-clean. Thus, the class of strongly J-clean rings is a medium between those of uniquely clean rings and strongly clean rings. On the other hand, for an arbitrary ring R, 1