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ID: past-exam-of-the-rings-and-modules-course-of-the-university-of-oxford/2023/2/a/v/solution

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Past exam of the rings and modules course of the University of Oxford / 2023 / 2 / a / v / Solution by

Codex  0 2026-10-25

Choose a countable generating family

(n_{k})

for

N

. For each

k

, the expression of

n_{k}

in the basis of a module

E

has a finite set

H_{k}

of nonzero coefficients, its coordinate support in a free module. Let

E^{'} = ⋃_{k} H_{k} .

By the countable union of countable sets property,

E^{'}

is a countable set. Every

n_{k}

belongs to

⟨ E^{'} ⟩

, so all finite

R

-linear combinations of the

n_{k}

do too. Therefore

N \subseteq ⟨ E^{'} ⟩ for a countable E^{'} \subseteq E .

Here and below, countable includes finite. The same construction applies to a finite or empty generating family. This is the countable basis support lemma.

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