Proposition 3

a) A semisimple module is without radical.

b) A module is semisimple and finitely generated if and only if it is without radical and Artinian.

By definition, the radical of a simple module is reduced to 0. If the module M is semisimple, then it is the direct sum of a family $(S_i)_{i\in I}$ of simple submodules, and we have $\mathfrak{R}(M) =\bigoplus_{i\in I}\mathfrak{R}(S_i)$ by Corollary 2 of Proposition 1 above, hence $\mathfrak{R}(M) = 0$.

If, moreover, M is finitely generated, then it is Artinian by Proposition 10 of VIII, p. 71.

Conversely, suppose that M is Artinian and without radical. By VIII, p. 2 applied to the set of maximal submodules of M, there exists a finite family $(N_i)_{i\in I}$ of maximal submodules of M whose intersection is reduced to 0. Then M is isomorphic to a submodule of $\bigoplus_{i\in I}(M/N_i)$. Therefore, it is semisimple and has finite length, and is a fortiori finitely generated.