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I have the following small insecurity:

Considering the infinite general linear group $GL(R)$ of some commutative ring $R$ with 1 Whitehead (p. 4 Theorem 1) and wiki/k-theory tells us that the commutator group is $E(R)=[GL(R),GL(R)]$ generated by elementary matrices.

Now wikipedia/elementary_matrix tells me also matrices with an element $r \neq 1$ on the diagonal is an elementary matrix (called row-multiplication matrices).

Can someone please verify that in our commutator group we just consider $E(R)=<\{M \ | \ M=I+rE_{i,j}, i\neq j, r \in R\}>$ i.e. the group generated by elementary matrices except of these row-scalar-multiplication matrices/ generated by elementary matrices with diagonal-entries 1 or do we also need those as generators?

Following my senses and this answer it should be only elementary matrices with diagonal-entries 1.

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Diagonal matrices with finite support and determinant 1 are generated by elementary matrices. This can be seen as follows: for $i\neq j$, define $e_{ij}(r)=I+rE_{ij}$, and for $r$ invertible, define $q_{ij}(r)=e_{ji}(r)e_{ij}(-r^{-1})e_{ji}(r)$. Then $q(r)=q_{12}(r)=\begin{pmatrix}0 & -r^{-1}\\ r & 0\end{pmatrix}$. So $q(r)q(-1)$ is the matrix $\begin{pmatrix}r^{-1} & 0\\ 0 & r\end{pmatrix}$. Thus $q_{ij}(r)q_{ij}(-1)$ is the diagonal matrix whose diagonal is 1 except $r^{-1}$ at $ii$ and $r$ at $jj$. From this it is immediate to conclude.

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