Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $R$ be a ring. An elementary matrix over $R$ is a matrix with $1$s along the diagonal and at most one other nonzero entry. Let $\text{EL}_n(R)$ denote the subgroup of $\text{GL}_n(R)$ generated by the elementary matrices.

I understand that $\text{EL}_n(R) = \text{SL}_n(R)$ provided that $R$ is a Euclidean domain.

Why does $\text{EL}_n(\mathbf{Z}[X_1,\ldots,X_m]) = \text{SL}_n(\mathbf{Z}[X_1,\ldots,X_m])$?

share|improve this question
Don't you also need to add $n\geq 3$? –  Gjergji Zaimi Feb 29 '12 at 0:47
Yes I think so. –  Sean Eberhard Mar 1 '12 at 11:18

1 Answer 1

up vote 6 down vote accepted

This is the "Suslin stability theorem". There is an algorithmic proof in Park+ Woodburn, "An algorithmic proof of the Suslin stability theorem for polynomial rings."

share|improve this answer
As Gjergji Zaimi notes in his comment, Suslin's theorem does require the assumption $n \geq 3$. –  Jim Humphreys Mar 1 '12 at 0:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.