short exact sequence of modules

模的短正合列

Algebra & Algebraic Geometry · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A sequence $0\to A\xrightarrow{i}B\xrightarrow{p}C\to0$ with $i$ injective, $p$ surjective and $\operatorname{im}i=\ker p$.

首个非零箭头单射、末个非零箭头满射且中间处像等于核的模序列。

Example

$0\to\mathbb Z\xrightarrow{2}\mathbb Z\to\mathbb Z/2\to0$ is exact.

Usage and distinctions

Exactness does not imply splitting; an $R$-linear section is additional data.

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Definition sources

1. Wen-Wei Li — Methods in Algebra I, author source正合列入门 (sec:exactseq-module), definition of 短正合列 before prop:split-ses-mod and following injective/surjective/kernel-image explanation · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷一，作者源文件正合列入门 (sec:exactseq-module), definition of 短正合列 before prop:split-ses-mod and following injective/surjective/kernel-image explanation · Checked 2026-10-15

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