\begin{definition}[Definition:Matrix Entrywise Addition/Ring]
Let $\struct {R, +, \cdot}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$.
Then the '''matrix entrywise sum of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A + \mathbf B$, and is defined as follows.
Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$.
Then:
:$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} + b_{i j}$
Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the (ring) addition operation $+$ on corresponding entries of $\mathbf A$ and $\mathbf B$.
That is, the '''matrix entrywise sum of $\mathbf A$ and $\mathbf B$''' is the '''Hadamard product''' of $\mathbf A$ and $\mathbf B$ with respect to ring addition.
This operation is called '''matrix entrywise addition'''.
\end{definition}