 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ib/paper-4/9e/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

P

be free with basis

(e_{i})_{i \in I}

. Given a surjection

f : M \to N

and a map

g : P \to N

, choose

m_{i} \in M

satisfying

f (m_{i}) = g (e_{i})

. There is a unique linear map

h : P \to M

with

h (e_{i}) = m_{i}

, and on every basis vector

(f \circ h) (e_{i}) = g (e_{i}) .

Thus $f \circ h = g$ , proving that free modules are projective.

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