Factorizations of Matrices Over Projective-free Rings

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