Unit_Matrix_is_Unity_of_Ring_of_Square_Matrices

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. The unit matrix over $R$ $\mathbf I_n = \begin {pmatrix} 1_R & 0_R & 0_R & \cdots & 0_R \\ 0_R & 1_R & 0_R & \cdots & 0_R \\ 0_R & 0_R & 1_R & \cdots & 0_R \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0_R & 0_R & 0_R & \cdots & 1_R \end {pmatrix}$ is the identity element of $\struct {\map {\MM_R} n, +, \times}$.