Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 2E / iii

Table of contents
- Solution iii

Solution

Change the generators to

a = m_{1}

,

b = m_{3} - m_{2}

and

c = m_{2}

. This is an invertible integer change of generators, since

m_{1} = a

,

m_{2} = c

,

m_{3} = b + c

. The relations become

2 b = 0

and

4 c = 0

, with no relation involving

a

. Consequently

M ≅ Z \oplus Z /2 Z \oplus Z /4 Z .

This is already the invariant-factor order required by the structure theorem for finitely generated modules over a principal ideal domain, because

2 ∣ 4

. The inverse change of generators shows that there are no hidden extra relations.

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