Example 4

For any two-sided ideal $\mathfrak{a}$ of A, denote the subset of $\mathbf{M}_n(A)$ consisting of the matrices with entries in $\mathfrak{a}$ by $\mathbf{M}_n(\mathfrak{a})$. It is a two-sided ideal of $\mathbf{M}_n(A)$. We have $\mathbf{M}_n(\mathfrak{a})A^n=\mathfrak{a}^n= A^n\mathfrak{a}$. It follows from Proposition 6 that every two-sided ideal of $\mathbf{M}_n(A)$ is of the form $\mathbf{M}_n(\mathfrak{a})$, where $\mathfrak{a}$ is a two-sided ideal of A.