Definition:Exact Sequence of Modules

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


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

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

Then the sequence $(1)$ is exact if and only if:

$\forall i: \operatorname{Im} d_i = \ker d_{i + 1}$

where $\operatorname{Im}$ and $\ker$ denote the image and kernel of mappings respectively.

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