In surveying LMR of T.Y.Lam and get the divisible module (for any ring with unity not necessary an integral domain) definition as follows:
``A right $R$-module $I_R$ is called divisible if and only if for any $u\in I_R$ and $a\in R$ such that $\mbox{ann}_r(a)\subseteq \mbox{ann}(u)$, $u=va$ for some $v\in I_R$.''
I'm having some dificulties to derive that "any factor module of a divisible module is also divisible''. To reach that I have the following equivalences:
$I_R$ is divisible $\Leftrightarrow \mbox{ann}^{I_R}(\mbox{ann}_l(a))=Ia\Leftrightarrow I$ is principally injective (i.e. for any $f\in\mbox{Hom}_R(aR,I)$ can be extended to $R$)
My attemp is so short:
Suppose that $I$ are divisible, $J$ submodule of $I$. Let $a\in R$, $\overline{u}\in I/J$ such that $\mbox{ann}_l(a)\subseteq \mbox{ann}(\overline{u})$ thus for all $r\in \mbox{ann}_l(a)$ we have $\overline{u}r=\overline{0}$, this implies that $ur\in J$.
At this point I'm freeze, i don't know how to get that $\mbox{ann}_r(a)\subseteq \mbox{ann}(m)$ or any fact that makes possible reach the result. Any help, hint or reference?