Definition:Zero Matrix/General Monoid

Definition
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 $S$.

The zero matrix of $\mathcal M_S \left({m, n}\right)$ is the $m \times n$ matrix whose elements are all $e$, 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 $\mathbf 0 = \left[{0}\right]_{m n}$.

Also see

 * Zero Matrix is Identity for Matrix Entrywise Addition
 * Zero Row or Column