2010-31.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	31	IB 2010 Groups, Rings and Modules	Paper 4, Section I, H	2010

Let $M$ be a free $\mathbb{Z}$-module generated by $e_{1}$ and $e_{2}$. Let $a, b$ be two non-zero integers, and $N$ be the submodule of $M$ generated by $a e_{1}+b e_{2}$. Prove that the quotient module $M / N$ is free if and only if $a, b$ are coprime.