Proposition 8

a) Let $V'$ and $V''$ be submodules of V such that $V''$ contains $V'$. The B-module $V''/V'$ is simple if and only if the A-module $\varphi (V'')/\varphi (V')$ is simple.

b) If $V'$ is a simple submodule, maximal submodule, or direct factor of V, then $\varphi (V')$ is, respectively, a simple submodule, maximal submodule, or direct factor of M.

c) The isomorphism $\varphi$ transforms the socle $\mathscr{S}(V)$ of V into the socle $\mathscr{S}(M)$ of $M*$and the radical (VIII, p. 151, Definition 1)$\mathfrak{R}(V)$ of V into the radical $\mathfrak{R}(M)$ of M$.*$

d) Let $(V_i)_{0\leqslant i\leqslant n}$ be a finite sequence of submodules of V. It is a Jordan– Hölder series of V if and only if $(\varphi (V_i)_{0\leqslant i\leqslant n})$ is a Jordan–Hölder series of M.