Definition:Short Exact Sequence of Modules

Definition
Let $\left({R, +, \cdot}\right)$ be a ring.

Let:
 * $(1): \quad \cdots \longrightarrow M_i \stackrel{d_i}{\longrightarrow} M_{i+1} \stackrel{d_{i+1}}{\longrightarrow} M_{i+2} \stackrel{d_{i+2}}{\longrightarrow} \cdots$

be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$.

Then the sequence $(1)$ is a short exact sequence if it exact, and is finite of the form


 * $ 0 \longrightarrow M_2 \stackrel{d_2}{\longrightarrow} M_3 \stackrel{d_3}{\longrightarrow} M_4 \longrightarrow 0$