\begin{definition}[Definition:Short Exact Sequence of Modules]
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
::$0 \longrightarrow M_2 \stackrel {d_2} {\longrightarrow} M_3 \stackrel {d_3} {\longrightarrow} M_4 \longrightarrow 0$
\end{definition}