Definition:Exact Sequence of Modules

Definition
Let


 * $ \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 \qquad (1)$

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


 * The sequence $(1)$ is null if $d_i \circ d_{i+1} = 0$ for all $i$.


 * The sequence $(1)$ is exact if $\operatorname{im} d_i = \ker d_{i+1}$ for all $i$, where $\operatorname{im}$ and $\ker$ denote the image and kernel of mappings repectively.


 * The sequence $(1)$ is short if it is finite and has the form


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