A class $S$ is closed under extension if given an ideal $I \subseteq R$ such that $I\in S$ and $R/I\in S$, then $R\in S$. A ring $R$ is left serial if it is a direct sum of left uniserial rings. Prove that the class $$S_R:=\{R:R~\text{is left serial composed of only finite left uniserial rings}\}$$ is closed under extensions.
We call a ring left uniserial finite length of if it has a unique composition series of finite length, that is, $R \supset L_1\supset L_2\supset\cdots\supset L_m = 0$ is the lattice of left ideals of $R$.