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(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.

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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.