Module theory MOC

Module homomorphism

Let 𝑅 be a ring and 𝑉,𝑊 be (left) 𝑅-modules. A map 𝑓 :𝑉 →𝑊 is a (left) 𝑅-module homomorphism or (left) 𝑅-linear iff for any 𝜆,𝜇 ∈ℝ and 𝑢,𝑣 ∈𝑉 module

𝑓(𝜆𝑢+𝜇𝑣)=𝜆𝑓(𝑢)+𝜇𝑓(𝑣)

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

Properties

• A linear map 𝑓 ∈𝑅𝖬𝗈𝖽(𝑉,𝑊) is epic iff it is surjective iff im⁡𝑓 =𝑊
• A linear map 𝑓 ∈𝑅𝖬𝗈𝖽(𝑉,𝑊) is monic iff it is injective iff ker⁡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 𝑅𝖬𝗈𝖽(𝑉,𝑉) is an 𝑅-algebra called the Endomorphism ring.

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