While reading an article about Sylvester matrix rank functions (generalization of the rank of a matrix for arbitrary rings) I needed to study the concept of inner rank, which is defined as follows:
Let $R$ be a ring and let $A$ be an $n\times m$ matrix over $R$. The inner rank of $A$, denoted as $\rho(A)$, is defined to be the least integer $k$ such that there exists two matrices $B$ and $C$, with dimensions $n\times k$ and $k\times m$ respectively, such that $A=BC$.
One of the properties the inner rank satisfies is the following:
$\rho\left(\begin{matrix} A \\ B \end{matrix}\right)\geq\max\lbrace\rho(A),\rho(B)\rbrace$ and $\rho\left(\begin{matrix} A & C \end{matrix}\right)\geq\max\lbrace\rho(A),\rho(C)\rbrace$ for any matrices $A,B$ and $C$ of proper size.
This was the only property I was not able to prove, since I don't know how to decompose a block matrix of this form into a product of two matrices, the other properties where easier since there was an obious decomposition of the matrix, but here either there is a more sophisticated argument or there is also an obvious decomposition that I can't see.
Any hint will be thanked.