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Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.

Under which conditions is the group $SL_n(R)$ generated by transvections?

(A transvection is a matrix with $1$ everywhere on the diagonal and exactly one other non-zero entry.) This is certainly the case if $R$ is a field, or if $R$ is a Euclidean domain, but I'm wondering whether there is a complete answer to the question.

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Just a remark, this is related to the special Whitehead group in the stable limit. – Ian Agol Mar 28 '11 at 19:52
I think this subject is covered in chapter 4 (probably section 4.3B) of "The classical groups and K-theory" by A. J. Hahn and O. T. O'Meara. Unfortunately I don't have my copy at hand – Max Horn Mar 28 '11 at 19:58
As Max says, the 1989 Springer treatise by Hahn-O'Meara contains a lot of information and references. The book is somewhat intimidating, partly because it deals with general rings, but the survey in section 4.3B is helpful for your question. These questions about generation of linear groups have a long history, going back to Dieudonne and before, with an active Russian school as well. There are few easy answers. – Jim Humphreys Mar 28 '11 at 20:11
The following paper may help: Bass, H. $K$-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. No. 22 1964 5--60. – Luis H Gallardo Mar 28 '11 at 22:13
P.S. To amplify my last remark, "complete answer" asks for quite a lot here. – Jim Humphreys Mar 29 '11 at 22:09
up vote 4 down vote accepted

I'm answering my own question based on the excellent reference given by Max and the additional comments of Jim Humphreys. There is nothing new in my answer, but I think it's useful to close the question in this way.

Following Hahn-O'Meara, we write $E_n(R)$ for the subgroup of $SL_n(R)$ generated by transvections (also called elementary matrices).

Theorem [H-O'M, Thm 4.3.9]. Let $R$ be a commutative ring. If $R$ is a Euclidean domain or a semilocal ring, then $SL_n(R) = E_n(R)$ for all $n$; If $R$ is a Hasse domain of a global field, then $SL_n(R) = E_n(R)$ for all $n \geq 3$ (and in many cases, but not always, also for $n=2$).

There are some other more general results known based on the so-called stable rank of the ring $R$, but as Jim pointed out, it seems hopeless to find a complete answer to the question.

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One of the important and useful result is that the above equality of groups hold for $n\geq 3$ for polynomial rings over a field. This is a result due to Suslin. – Mohan Apr 4 '11 at 20:55

Further results are known: L. Vaserstein's paper "SL_2 of Dedekind rings of arithmetic type" proves these rings are generalized euclidean when they have a unit of infinite order. Integral group rings of finite groups are generalized euclidean when the group has no homomorphic image among the generalized quaternion groups of order a multiple of 4, no image among the binary polyhedral groups, and the abelianization of the group has generalized euclidean integral group ring. The finite abelian G with ZG euclidean include the cyclic groups, and Z/2 x Z/2, by the 1984 paper "Generalized euclidean group rings" by Dennis, Magurn & Vaserstein. But ZG is not generalized euclidean when SK_1(Z[G/[G,G]]) is non-vanishing, as it is for Z/4 x Z/2 x Z/2, for instance. So this is a delicate property!

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The goal of my answer is only to provide recent references. I warmly recommend these two bits of T. Y Lam's book [1]:

  • §I.8, for examples where transvections fail to generate $SL_n(R)$
  • the second to last paragraph of §VIII.12 for other interesting examples of rings $R$ satisfying $SL_2(R) = E_2(R)$ or its negative.

And also B. Magurn's latest article on generalized Euclidean group rings [2].

[1] Serre's problem on projective modules, T. Y. Lam, 2006
[2] On a note from Oliver concerning generalized Euclidean group rings. B. Magurn, 2014

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A nice account of the case

$$ n =2 $$

is given by I. Reiner in his review of a paper of P.M. Cohn below

The review is very detailed, Hope the tex may compile...

(that NOT worked !)

I give then just the review to try in MR:

MR0207856 (34 #7670) Cohn, P. M. On the structure of the ${\rm GL}_{2}$ of a ring. Inst. Hautes Études Sci. Publ. Math. No. 30 1966 5–53. 20.70 (16.48)

and the beginning of the review:

This well-written article encompasses a wealth of information about general linear groups over certain classes of rings. The author generalizes many earlier results about such groups, and gives a number of new and striking results. We proceed to describe some of the main theorems. Assume throughout that the underlying ring $R$ has a unity element and is associative, though not necessarily commutative. Denote by $U(R)$ its groups of units. (1) Let $\text{GL}_n(R)$ be the group of $n\times n$ invertible matrices over $R$, and $D_n(R)$ its subgroup of diagonal matrices. Let $E_n(R)$ be the group generated by the set of transvections $\{I+ae_{ij}\colon a\in R,1\leq i,j\leq n,i\neq j\}$, where $\{e_{ij}\}$ is a set of matrix units. Define $\text{GE}_n(R)=D_n(R)\cdot E_n(R)$, the subgroup of $\text{GL}_n(R)$ generated by elementary matrices. Of course, $E_n(R)\Delta\text{GE}_n(R)$. The author calls $R$ a generalized Euclidean ring (GE-ring) if $\text{GL}_n(R)=\text{GE}_n(R)$ for all $n$.

$$ \dots $$

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The original review's typo (or, at least, the not-so-common delta notation) is faithfully reproduced in the second to last line above. I. Reiner probably intended to say that $E_n(R)$ is a normal subgroup of $GE_n(R)$. – Luc Guyot Jun 4 at 17:44

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