Example 1

Let $\mathfrak{a}$ be a left ideal of A. The radical of the A-module $A_s/\mathfrak{a}$ is equal to $\mathfrak{a}'/\mathfrak{a}$, where $\mathfrak{a}'$ is the intersection of the maximal left ideals of A containing $\mathfrak{a}$. In particular, the radical of the $\mathbf{Z}$-module $\mathbf{Z}$ is reduced to 0, and that of the $\mathbf{Z}$-module $\mathbf{Z}/p^n\mathbf{Z}$ is equal to $p\mathbf{Z}/p^n\mathbf{Z}$ for every prime number $p$ and every integer $n\geqslant 1$.