Module

Finitely generated module

An 𝑅-module 𝑀 is said to be finitely generated iff it has a finite spanning set {𝑚𝑖}𝑟𝑖=1 module so that

𝑀=span𝑅⁡{𝑚𝑖}𝑟𝑖=1

Thus a vector space is finitely generate iff it is finite dimensional, however the situation is more complicated over a general ring.

Properties

1. Let 𝑀 ∈𝑅𝖬𝗈𝖽, 𝑁 ≤𝑅𝑀. If 𝑁 and 𝑀/𝑁 are finitely generated, then so too is 𝑀.

Proof of 1.

Suppose 𝑀/𝑁 =⟨𝑚1 +𝑁,…,𝑚𝑟 +𝑁⟩𝑅 and 𝑁 =⟨𝑛1,…,𝑛𝑡⟩𝑅. Then

𝑀=⟨𝑚1,…,𝑚𝑟,𝑛1,…,𝑛𝑡⟩𝑅

proving ^P1.

Other results

• Finitely generated module over a module-finite R-ring
• Finitely generated modules over a noetherian ring are noetherian (^P2)

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