Definition: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 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.