Factorizations of Matrices Over Projective-free Rings
  An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings. 
H. Chen, H. Kose, Y. Kurtulmaz
math.RA