Let $R$ be a ring. An $m\times n$-matrix in $R$ ($m$, $n\in\N^+$) is a function from $\llbra m\rrbra\times\llbra n\rrbra$ to $R$.
The set of all $m\times n$-matrices in $R$ is
\[\mat_R(m,n) = \fun(\llbra m\rrbra\times\llbra n\rrbra,R).\]An $m\times n$-matrix $M$ can be explicitly written as
\[M = \begin{bmatrix} M(1,1) & \cdots & M(1,n) \\ \vdots & \ddots & \vdots \\ M(m,1) & \cdots & M(m,n) \end{bmatrix}.\]
Let $R$ be a ring and $A$, $B\in\mat_R(m,n)$ ($m$, $n\in\N^+$). The sum of $A$ and $B$ is
\[A + B \in \mat_R(m,n) : (i,j) \mapsto A(i,j) + B(i,j).\]
Let $R$ be a ring, $A\in\mat_R(m,n)$ and $B\in\mat_R(n,p)$ ($m$, $n\in\N^+$). The product of $A$ and $B$ is
\[AB \in \mat_R(m,p) : (i,j) \mapsto \sum_{k=1}^n A(i,k)B(k,j).\]