It is a homework asking me to prove the Smith Normal Form theorem, stating that
If $R$ is a PID, and $A$ is a $n\times n$ matrix over $R$, then there exists invertible matrices $P, Q$ such that $PAQ$ is diagonal.
There is a hint at the end of the question, saying that let $F=R^n$ and $\alpha$ be the module homomorphism from $F$ to $F$ with the matrix $A$. First we show $F=U\oplus\ker(\alpha)$ for some $U\cong R^m$ with $m\leqslant n$. I have done this.
The hint continues and says we then find a basis of $\{u_1,\cdots,u_m\}$ for $U$, $\{u_{m+1},\cdots,u_n\}$ for $\ker(\alpha)$, and $\{v_1,\cdots,v_n\}$ for $F$, such that $\alpha$ is represented as a diagonal matrix w.r.t. basis $\{u_1,\cdots,u_n\}$ and $\{v_1,\cdots,v_n\}$.
However, I cannot construct such basis. Can anyone help?
Update:
R is a commutative ring, because it is the homework of the course Commutative Algebra.