Does the solvability of an equation $I=MV+NC$ for $V$ depend on $Ker(C)$?

Question

In a recently read paper I came across the statement that for matrices $M\in\Re^{n\times{n}_{1}}$, $N\in\Re^{n\times{s}_1}$, $C\in\Re^{s_1\times{n}}$, the equation
\begin{equation} I=MV+NC \tag{1} \end{equation} has a solution for $V\in\Re^{n_1\times{n}}$ if the number of linear independent rows of $V$ must be at least equal to the dimension of $Ker(C)$. To show that the claim is true, lets take a vector $x\in{Ker(C)}$, then the above equation reduces to \begin{equation} x=MVx=M\begin{bmatrix}V_{1}x\\V_{2}x\\\vdots\\V_{n_1}x\end{bmatrix} \tag{2} \end{equation} where $V_{i}$ is the $i^{th}$ row of $V$. Since $Ker(C)\neq\{0\}$ and thus $x\neq{0}$ for which equation 2 holds if $MV=I$ or $rank(MV)\leq{n}$. But I do not see the way forward. Any hint is greatly appreciated.