I'm a little bit puzzled. So Wikipedia says that the Smith normal form applies to matrices with entries in a principal ideal domain such as $\mathbb Z$ or $F[x]$. However, in the algebra books I can only find forms of the theorem for $F[x]$, but what I need is a reference for $\mathbb Z$. So, is $\mathbb Z$ isomorphic to some polynomial ring $F[x]$? I've tried with $F = \mathbb F_2$, but that doesn't seem to work.
Sorry for this stupid question, but I'm very unexperienced with algebra.