Let $$\mathbf{A}= \begin{bmatrix} A_{11} & A_{12}&\dots &A_{1r} \\ A_{21} &A_{22} &\dots &A_{2r} \\ \vdots & \vdots &\ddots &\vdots \\ A_{s1} &A_{s2} &\dots &A_{sr} \end{bmatrix}$$ be an $n\times m$ matrix on a ring. where the $A_{ij}$ are sub-matrices (rectangular blocks of entries). Let $$\mathbf{B}= \begin{bmatrix} B_{11} & B_{12}&\dots &B_{1q} \\ B_{21} &B_{22} &\dots &B_{2q} \\ \vdots & \vdots &\ddots &\vdots \\ B_{r1} &B_{r2} &\dots &B_{rq} \end{bmatrix}$$ be a similar $m\times p$ matrix on the same ring.
Assuming $$\sum_{k=1}^{r} \begin{bmatrix} A_{1k}B_{k1} & A_{1k}B_{k2}&\dots &A_{1k}B_{kq} \\ A_{2k}B_{k1} &A_{2k}B_{k2} &\dots &A_{2k}B_{kq} \\ \vdots & \vdots &\ddots &\vdots \\ A_{sk}B_{k1} &A_{sk}B_{k2} &\dots &A_{sk}B_{kq} \end{bmatrix}$$ is a valid expression (including valid matrix mutiplications), is it equal to $\mathbf{AB}$?
As an example is this a valid mutiplication of $3\times 3$ matrices?:
Yes. This follows from the definition of matrix multiplication. In fact, the subblocks don't need to be square. As long as they have conforming sizes, you can evaluate $AB$ blockwise.