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