Module homomorphism

Let $𝑅$ be a ring and $𝑉, 𝑊$ be (left) $𝑅$ -modules. A map $𝑓 : 𝑉 \to 𝑊$ is a (left) $𝑅$ -module homomorphism or (left) $𝑅$ -linear iff for any $𝜆, 𝜇 \in ℝ$ and $𝑢, 𝑣 \in 𝑉$ #m/def/module

𝑓 (𝜆 𝑢 + 𝜇 𝑣) = 𝜆 𝑓 (𝑢) + 𝜇 𝑓 (𝑣)

This is a direct generalization of a linear map between vector spaces.

Properties

A linear map $𝑓 \in 𝑅 𝖬 𝗈 𝖽 (𝑉, 𝑊)$ is epic iff it is surjective iff $i m 𝑓 = 𝑊$
A linear map $𝑓 \in 𝑅 𝖬 𝗈 𝖽 (𝑉, 𝑊)$ is monic iff it is injective iff $k e r 𝑓 = {0}$
A linear map is an isomorphism iff it is bijective iff it is epic and monic
If $𝑅$ is a commutative ring, then $𝑅 𝖬 𝗈 𝖽 (𝑉, 𝑉)$ is an $𝑅$ -algebra called the Endomorphism ring.

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