Proposition 2

Let A be a left Artinian ring. For every A-module M, the following properties are equivalent:

(i) M is semisimple.

(ii) M is without radical.

(iii) M is annihilated by the radical $\mathfrak{r}$ of A.

We know that (i) implies (ii) (VIII, p. 153, Proposition 3). On the other hand, $\mathfrak{r}M$ is contained in the radical of M (VIII, p. 158, Proposition 6), so (ii) implies (iii).

Suppose that the A-module M is annihilated by $\mathfrak{r}$. We can view it as a module over the ring $A/\mathfrak{r}$. Now, the ring $A/\mathfrak{r}$ is semisimple (VIII, p. 173, Proposition 1), and every module over a semisimple ring is semisimple (VIII, p. 138, Proposition 4). Consequently, M is a semisimple module over the ring $A/\mathfrak{r}$ and a fortiori over the ring A. Hence (iii) implies (i).