Strongly J-clean matrices over 2-projective-free rings
  An element $a$ in a ring $R$ 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 $2\times 2$ matrices over 2-projective-free non-commutative rings. 
Marjan Sheibani Abdolyousefi, Hunyin Chen, Rahman Bahmani Sangesari
math.RA