Definition:Short Exact Sequence of Modules

Definition

Let $\struct {R, +, \cdot}$ 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 is exact, and is finite of the form:

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

Also see

• Definition:Exact Sequence of Modules

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