Quotient module

Given a Module $𝑀 \in 𝑅 𝖬 𝗈 𝖽$ and a Submodule $𝑁 \leq 𝑀$ , the quotient module $𝑀 / 𝑁$ is the quotient group with a natural $𝑅$ -action: #m/def/module

𝑟 \cdot (𝑣 + 𝑁) = 𝑟 \cdot 𝑣 + 𝑁

for any $𝑟 \in 𝑅$ and $𝑣 \in 𝑀$ .

More explicitly

$𝑀 / 𝑁$ is just $𝑀$ with all elements of $𝑁$ collapsed to zero. More formally, using the congruence relation

𝑥 \equiv 𝑦 ⟺ 𝑥 - 𝑦 \in 𝑁

we have $𝑀 / 𝑁 = 𝑀 / \equiv$ with $𝛼 [𝑎] + 𝛽 [𝑏] = [𝛼 𝑎 + 𝛽 𝑏]$ .

We thus have the short exact sequence in $𝑅 𝖬 𝗈 𝖽$

0 \to 𝑁 ↪ 𝑀 𝜋 ↠ 𝑀 / 𝑁 \to 0

Universal property

The quotient module with the canonical projection $(𝑀 / 𝑁, 𝜋)$ is characterised up to unique isomorphism by the universal property:

$𝑁 \subseteq k e r 𝜋$ . If $𝑁 \in 𝑅 𝖬 𝗈 𝖽$ is a module and $𝜑 \in 𝑅 𝖬 𝗈 𝖽 (𝑀, 𝑁)$ is a morphism with $𝑆 \in k e r 𝜑$ , then there exists a unique morphism $―― 𝜑 \in 𝑅 𝖬 𝗈 𝖽 (𝑀 / 𝑆, 𝑁)$ so that $𝜑 = ―― 𝜑 𝜋$ .

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