Finitely generated module

An $𝑅$ -module $𝑀$ is said to be finitely generated iff it has a finite spanning set ${𝑚 𝑖} 𝑟 𝑖 = 1$ #m/def/module so that

𝑀 = s p a n 𝑅 {𝑚 𝑖} 𝑟 𝑖 = 1

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

Properties

Let $𝑀 \in 𝑅 𝖬 𝗈 𝖽$ , $𝑁 \leq 𝑅 𝑀$ . If $𝑁$ and $𝑀 / 𝑁$ are finitely generated, then so too is $𝑀$ .

Proof of 1.

Suppose $𝑀 / 𝑁 = ⟨ 𝑚 1 + 𝑁, \dots, 𝑚 𝑟 + 𝑁 ⟩ 𝑅$ and $𝑁 = ⟨ 𝑛 1, \dots, 𝑛 𝑡 ⟩ 𝑅$ . Then

𝑀 = ⟨ 𝑚 1, \dots, 𝑚 𝑟, 𝑛 1, \dots, 𝑛 𝑡 ⟩ 𝑅

proving ^P1.

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