Invertibility of structured perturbations of singular matrices over unital rings

In this article, we study the invertibility of matrices of the form $A+EDF$, where $A$ and $D$ are square matrices over a unital, not necessarily commutative, ring and $A$ is singular. Under natural hypotheses on kernels and images, we prove that $A+EDF$ is invertible if and only if $D$ is invertibl