Let $R$ be a ring and $\mathcal{C}$ be the category of left semi-simple $R-$modules.
$\textbf{Q:}$ What is the example of $\mathcal{C}$ is not closed under extension?(i.e. Given $A,B$ semi-simple modules, there is $X$ not a semi-simple module s.t. $0\to A\to C\to B\to 0$ is $R$ left module exact sequence.) I have thought of $Z_6$ over $Z_6$. $Z_6\cong Z_2\times Z_3$. Now $0\to Z_2\to Z_2\times Z_3\to Z_3\to 0$ is an extension of $Z_2,Z_3$ as $Z_6$ module. I do not see other obvious possible extensions.
Ref. Auslander, Reiten, Representation Theory of Artin Algebras.