Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 1 / b / i

Table of contents
- Solution i

Solution

Let

n_{1}, \dots, n_{r}

generate

N

, and let the images of

m_{1}, \dots, m_{s}

generate

M / N

. For any

m \in M

, subtracting a suitable linear combination of the

m_{j}

leaves an element of

N

, which is a combination of the

n_{i}

. Thus

M = R n_{1} + \dots + R n_{r} + R m_{1} + \dots + R m_{s} .

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