Definition: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 $d_i$.

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: Specify the index set. Is it $\Z$ or $\N$, or may be finite? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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.

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: Clarify which sequence. I think exact sequence is not a sequence. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Definition:Short Exact Sequence of Modules

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