Definition:Exact Sequence of Modules

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Let $\struct {R, +, \cdot}$ 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:

$\forall i: \Img {d_i} = \map \ker {d_{i + 1} }$

where $\Img {d_i}$ and $\map \ker {d_{i + 1} }$ denote the image and kernel of homomorphisms respectively.

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