2006-8.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	8	IB 2006 Groups, Rings and Modules	2.I.2E	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 $\mathbb{Z}$-module $M$ with generators $\left{m_{1}, m_{2}, m_{3}\right}$ and relations $2 m_{3}=2 m_{2}, 4 m_{2}=0$.