Let $R$ be a ring with $1$ and $M$ a left $R-$module. The module $M$ is said to be uniserial if the poset of submodules of $M$ is a chain. Let $M$ be a uniserial module and $\alpha,\beta\in End_R(M)$.
Why we have: $\alpha\beta$ is injective if and only if both $\alpha$ and $\beta$ are injective.
Thanks in advance
One direction is obvious: if $\alpha, \beta$ are injective, so is $\alpha \beta$.
Hint for the other: "$\alpha \beta$ is injective" means in particular that $\beta$ is injective, so the question is whether $\alpha$ is injective as well. But look at the submodules $\mathrm{Ker}\, \alpha, \mathrm{Im}\, \beta:$ One has to have one in the other by uniseriality. Use this to show that we necessarily have $\mathrm{Ker}\, \alpha=0$.