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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. – Geoff Robinson Jan 15 '12 at 11:12
It seems you mean this article, whose title is "On the structure of the $GL_2$ of a ring"? – joriki Jan 15 '12 at 11:14
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. – joriki Jan 15 '12 at 11:19
@joriki : Yes, it was a typo. Thank you. – Giuseppe Negro Jan 15 '12 at 11:21
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. – joriki Jan 15 '12 at 11:23
up vote 8 down vote accepted

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.

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