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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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    I am not sure about this, but is it possible that the usual algorithm for reducing a matrix to Smith Normal Form requires a Euclidean domain? I know that the theory of finitely generated modules over a PID guarantees the existence of Smith Normal form, but that is an existence result, rather than a constructive one.2012-01-15
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    It seems you mean [this article](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1966__30_/PMIHES_1966__30__5_0/PMIHES_1966__30__5_0.pdf), whose title is "On the structure of *the* $GL_2$ *of* a ring"?2012-01-15
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    The Wikipedia article you cite uses multiplication with a full $2\times2$ matrix, so one explanation could be that this sometimes can't be written as the product of two elementary matrices.2012-01-15
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    @joriki : Yes, it was a typo. Thank you.2012-01-15
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    Indeed Section 2 of the article makes just that distinction, between the set of invertible $2\times2$ matrices and the set of $2\times2$ matrices generated by elementary matrices, and states that the two coincide for Euclidean rings.2012-01-15
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    @GeoffRobinson, joriki: I think you're right, I had missed this point. In the non-Euclidean case reduction to SNF requires some other matrix besides elementary ones. This leads to failure of my argument above. Thank you both! If you want to publish your comment as an answer I will accept it.2012-01-15

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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.