Matrix over commutative ring with noncommutative diagonal perturbation

Question

I'm interested in the inverse and determinant properties of a relatively small matrix $A+D$, where $A\in M_n(R_c)$ for a commutative ring $R_c$ and $D\in M_n(R_n)$ is a diagonal matrix over a non-commutative ring $R_d$. Are there any special properties for such a matrix? E.g., Manin matrices are non-commutative, but many of the classical linear algebra results hold. This is not Manin, but has a nice structure compared to the general non-commutative setting.