Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 9E / ii

Table of contents
- Solution ii

Solution

Assume (ii), and let an

R

M \subseteq F

be generated by

x_{1}, \dots, x_{m}

. Choose a common nonzero denominator

d

and write

x_{j} = a_{j} / d

with

a_{j} \in R

. Repeated application of (ii) shows that

(a_{1}, \dots, a_{m}) = (g)

for one element

g

: the gcd at each step is a linear combination of the elements processed so far. Therefore

M = R \frac{g}{d},

so (iii) holds. This is the finite-ideal property defining a Bézout domain.

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