(Cfr. Wikipedia for the definition of Elementary matrix).
Have a look at the following excerpt of Jacobson's Basic algebra vol.I, 2nd edition, pag.186.
There exist PID in which not every invertible matrix is a product of elementary ones. An example of this type is given in a paper by P.M.Cohn, On the structure of the $\text{GL}_2$ of a ring, Institut des Hautes Etudes Scientifiques, #30 (1966), pp 5 - 54.
This leaves me puzzled. Take an invertible matrix $A$ over a PID. Then $A$ has a Smith normal form, that is, up to elementary row and columns operations it is equivalent to something like this
$\begin{bmatrix} d_1 & && \\ & d_2 &&\\ &&\ddots&\\ &&&d_n\end{bmatrix}.$
In particular $\det A= d_1\ldots d_n u$ for some unit element $u$. But $\det A$ needs be unit, so all of $d_i$'s are units, which means that up to some other elementary row operation $A$ is equivalent to the identity matrix. It seems to me that we have just proven that $A$ is the product of elementary matrices, which is false as of Jacobson's claim.
There must be an error somewhere, but where?
Thank you.