This question was asked in my assignment of Commutative algebra and I am not able to make significant progress on it. I am following Algebra by Thomas Hungerford.
Let $R$ be a ring and let we have a commutative diagram of $R$-modules $\alpha_1 :M_1\to N_1$ and $\alpha_2 :M_2 \to N_2$, in which the two horizontal maps $M_1 \to M_2$ and $N_1 \to N_2$ are injective. Show that there exists an $R$-module $E$ and an exact sequence $$0 → \ker α_1 → \ker α_2 → E → \operatorname{coker} α_1 → \operatorname{coker} α_2 .$$ of $R$-modules.
I don't know which result should I use to move forward to prove what is asked.
Can you please help me?