Let $R$ be an D266: Identity ring such that

(i)	$0_R$ is an D270: Additive identity in $R$
(ii)	$1_R$ is a D577: Multiplicative identity in $R$

A D102: Matrix $r : I \times J \to R$ is an identity matrix over $R$ if and only if

(1)	\begin{equation} \forall \, i \in I : \forall \, j \in J \left( i \neq j \quad \implies \quad r_{i, j} = 0_R \right) \end{equation}
(2)	\begin{equation} \forall \, i \in I : \forall \, j \in J \left( i = j \quad \implies \quad r_{i, j} = 1_R \right) \end{equation}