\begin{definition}[Definition:Matrix Product (Conventional)/Pre-Multiplication]
Let $\struct {R, +, \circ}$ be a ring.
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.
Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$.
Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$.
Then $\mathbf B$ is '''pre-multiplied''' by $\mathbf A$.
\end{definition}