(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.
Best Answer
The argument fails because the reduction to Smith normal form may require a full $2\times2$ matrix that can't be written as a product of elementary matrices. The cited paper gives an example of such a $2\times2$ matrix over $\mathbb Q(\sqrt{-19})$ on page 23.