Definition:Zero Matrix/General Monoid

Theorem
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$.

Let $\mathcal M_S \left({m, n}\right)$ be an $m \times n$ matrix space over $S$.

Then $\left({\mathcal M_S \left({m, n}\right), +}\right)$ has an identity.

This identity element, called the zero matrix, has all elements equal to $e$, and can be written $\left[{e}\right]_{m n}$.

If the monoid $S$ is a number field in which the additive identity is represented as $0$, the zero matrix is usually written $\mathbf 0 = \left[{0}\right]_{m n}$.

Zero Row or Column
A row or column in which all the elements are equal to $e$ is called a zero row or zero column.

Proof
Let $\left[{a}\right]_{m n} \in \mathcal M_S \left({m, n}\right)$, where $\left({S, \circ}\right)$ is a monoid.

Let $a_{i j}$ be an element of $\left[{a}\right]_{m n}$.

Then $\forall \left({i, j}\right) \in \left[{1 \,.\,.\, m}\right] \times \left[{1 \,.\,.\, n}\right]: a_{i j} \circ e = a_{i j} = e \circ a_{i j}$.