Let $R$ be a Principal Ideal Domain. Let $M=\oplus_{i=1}^n R$ be a free module of rank $n$. Let $v=(a_{1},a_{2},...a_{n})\in M$ be such that ideal generated by $a_{1},a_{2},...a_{n}$ is $R$.
Show that:
$(a)$ $M/Rv$ is a free $R$ module,
$(b)$ There is a invetible matrix $A\in GL_{n}(R)$ whose first row is $v$.
Can anyone explain to me, what the question is trying to ask? and what it seeks to establish?
Moreover, please provide me a rough idea for the proof.