2.I.2E

Groups, Rings and Modules
Part IB, 2006

(i) Give the definition of a Euclidean domain and, with justification, an example of a Euclidean domain that is not a field.

(ii) State the structure theorem for finitely generated modules over a Euclidean domain.

(iii) In terms of your answer to (ii), describe the structure of the Z\mathbb{Z}-module MM with generators {m1,m2,m3}\left\{m_{1}, m_{2}, m_{3}\right\} and relations 2m3=2m2,4m2=02 m_{3}=2 m_{2}, 4 m_{2}=0.