If every matrix has a Smith normal form, then every finitely generated
$R$-submodule $M$ of $R^n$ satisfies $R^n/M$ is a finite
direct sum of modules isomorphic to $R/aR$. If $R$ is Noetherian
this implies that every finitely generated module is a direct sum of modules
of the form $R/aR$. So if $I$ is a maximal ideal of the Noetherian $R$
then $R/I$ is a simple ideal, so if $R/I\cong R/aR$ then $I=aR$ is
principal. So in a Noetherian ring with Smith normal form for all matrices, every
maximal ideal is principal. Does this imply that all ideals
are principal?....I'm not sure :-)