# Smith Normal Form for PIDs

Could someone provide a good reference to look up the existence and uniqueness of Smith Normal Form (SNF) for a PID?

I have seen it done for Euclidean domains but not for a PIDs. I know the difference is not much, but I would like a good reference. Thanks.

• The 3 elementary row and column operations are not enough to obtain SNF over a PID. – messi May 23 '13 at 9:40
• What messi says. If $\gcd(a,b)=1$, then Bezout says that we can find $u,v$ such that $au-bv=1$, and we need to include 2-D transforms of the form $$\pmatrix{a&b\cr v&u\cr}$$ in the list of operations. – Jyrki Lahtonen May 23 '13 at 21:29