The elementary divisor theorem was originally proved by a calculation on integer matrices, using elementary (invertible) row and column operations to put the matrix into Smith normal form. That is the matrix is zero off the diagonal, and on the diagonal each entry divides the one below it.

This calculation immediately generalizes to matrices over any Euclidean ring.

The key point is that in a Euclidean ring the GCD of two elements can be found by a series of steps where, in each step, one of the arguments is *not* multiplied by any non-unit. Specifically you reduce an entry $B$ modulo an entry $A$ by adding some (generally non-unit) multiple of A to B to get a result with smaller Euclidean norm than $A$. But $B$ is not multiplied by anything in this step, and so certainly not multiplied by any non-unit.

This does not generalize directly to every PID. In a PID elements $A,B$ have a GCD which is a linear combination of them, But that linear combination may require non-unit multiples of each (even in the integers). And it is not obvious that in every PID the process can be broken into steps where one or the other argument enters the linear combination with no multiplier (or at worst some unit multiplier).

Is there either some way to do it that I have not seen, or a proof that in some PIDs it cannot be done? Can one calculate Smith normal forms over PIDs by elementary matrix operations?

The motivation for this question is to understand exactly what facts Emmy Noether faced when she gave her abstract algebra proof of the elementary divisor theorem for PIDs. Her proof was one striking confirmation that the Noetherian condition on rings is a powerfully useful idea.

The discussion by მამუკა ჯიბლაძე and Luc Guyot reveals the interesting point that, while the answer to my question is simply no for the case of the integers in $\mathbb{Q}(\sqrt{-19})$, you actually can get Smith normal forms for that case by essentially the same kind of arithmetic, if you also allow yourself to also add new columns to the matrix as new `work space.' But algebraic $K$-theory shows the answer remains negative for other cases even when you allow that. As to the history, it seems Noether could well have seen elementary matrix calculations as hopeless for the problem, but she could not have proved they cannot work. She is not known to have ever done things like concrete calculations on $\sqrt{-19}$.

invariant factorsinstead. Elementary divisors are factors $p^m$ of the invariant factors, where $p$ is irreducible. Even in a Euclidian domain, one cannot always compute them explicitly. For instance in a polynomial ring, one cannot factorize explicitly most polynomials of degree $\ge5$. $\endgroup$