# Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity

Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $$\mathrm{GL}_n(\mathbb{Z})$$, $$\forall n\geq 0$$. I have understood their proof and tried to get a result when $$\mathbb{Z}$$ is replaced by a PID with unity $$R$$. Even for the $$n \leq 3$$ I could not find the commutator subgroups (of course for $$n =1$$, they are the units in the PID). The difficulty arises because of the existence of units which are not $$1$$ or $$-1$$.

I would be obliged if I get any help in this direction or some references, where my question already has an answer. Thank you.

• For $n\ge 3$, every elementary matrix is a commutator ($[e_{ij}(1),e_{jk}(s)]=e_{ik}(s)$) and $\mathrm{SL}_n$ of a PID is generated by elementary matrices, so the determinant yields the abelianization. So it remains only $n=2$ to consider. For $n=2$, when the PID is Euclidean then generation by elementary matrices also holds, but I'm not sure in general, and in any case the commutator formula does not apply.
– YCor
Apr 2 at 9:22
• Apr 2 at 9:42
• Yes, I made no claim for $n=2$. For a Euclidean PID, $\mathrm{SL}_2(R)$ is generated by elementary matrices, but as you mention with $R=\mathbf{Z}$, unipotents need not be in the derived subgroup.
– YCor
Apr 2 at 11:19
• So for $n\ge 1$ and a PID $R$, the determinant gives a surjective homomorphism from the abelianization of $\mathrm{GL}_n(R)$ to $R^\times$, whose kernel is trivial for $n\neq 2$. For $n=2$, when $R$ is also Euclidean, it still holds that the kernel is an elementary abelian 2-group (namely a quotient of $R/2R$).
– YCor
Apr 2 at 11:30
• If $R$ is a Euclidean domain and has an invertible element $s$ such that $1-s$ is invertible, then all elementary matrices are commutators. This applies to $R=k[t]$ whenever $k$ is a field with $|k|>2$. As regards $R=\mathbf{F}_2[t]$, the abelianization of $\mathrm{GL}_2(R)$ is a free $\mathbf{F}_2$-module of infinite rank (as seen from the usual amalgam decomposition).
– YCor
Apr 2 at 11:43

This answer provides references for the facts mentioned in the comments of YCor. More on this topic can be found in [2, Section 9].

For $$R$$ a unital ring, we denote by $$\text{E}_n(R)$$ the subgroup of $$\text{GL}_n(R)$$ generated by all the transvections $$e_{ij}(r) = I_n + r \epsilon_{ij}$$ (a.k.a the elementary matrices over $$R$$) with $$r \in R, 1\le i \neq j \le n$$ and where $$I_n$$ is identity matrix and $$\epsilon_{ij}$$ is the matrix whose $$(i,j)$$-entry is $$1$$ while all its other entries are zero.

Claim. Let $$R$$ be a unital ring of finite stable rank $$\text{sr(R)}$$. Then we have $$[\text{GL}_n(R), \text{GL}_n(R)] = \text{E}_n(R)$$ for every $$n > \text{sr}(R)$$.

Proof. This follows for instance from Vaserstein's Injective Stability Theorem [4, Theorem 10.15].

Corollary. Let $$R$$ be a commutative and unital ring of Krull dimension at most $$1$$, e.g., a principal ideal domain. Then $$[\text{GL}_n(R), \text{GL}_n(R)] = \text{E}_n(R)$$ for every $$n > 2$$.

Proof. As the stable rank of a commutative and unital ring of Krull dimension at most $$d$$, is at most $$d + 1$$ [3, Corollary 2.3], the result follows from the previous claim.

For $$n = 2$$, Section 9 of P. M. Cohn's [2] contains a wealth of valuable results.

For instance, it follows from [2, Proposition 9.2] that $$[\text{GL}_2(R), \text{GL}_2(R)] = \text{E}_2(R)$$ if $$R$$ is Euclidean and $$1$$ is the sum of two units of $$R$$, e.g., if $$R$$ is any of the following rings:

• $$\mathbb{Z}[\frac{1}{2n}]$$ where $$n \in \mathbb{N}_{> 0}$$,
• $$\mathbb{Z}_{(p)} = \{ \frac{m}{n} \, \vert \, m,n \in \mathbb{Z}, \text{gcd}(n, p) = 1 \}$$ for $$p$$ an odd prime number,
• $$\mathbb{Z}_p$$, the ring of $$p$$-adic integers for $$p$$ an odd prime number,
• the ring $$k[X]$$ of univariate polynomials over $$k$$ where $$k$$ is a field with at least $$3$$ elements; see how this result changes when $$k$$ is $$\mathbb{F}_2$$, the field with two elements, in this related post.
• $$\mathbb{Z}[e^{\frac{2i \pi}{3}}]$$, the ring of Eisenstein integers.

Among other interesting results, Cohn's generalization [2, Theorem 9.4 and subsequent remark] of the result of Hua and Reiner [1] is particularly relevant: we have $$\text{E}_2(R)/[\text{GL}_2(R), \text{GL}_2(R)] \simeq R/N$$ where $$N$$ is the ideal of $$R$$ generated by all the elements of the form $$1 - \alpha$$ with $$\alpha \in \text{GL}_1(R)$$, provided $$R$$ is quasi-free. Quasi-free rings in the sense of Cohn encompass the class of discretely normed rings which contains in particular $$\mathbb{Z}$$, the rings of rational integers and $$\mathbb{Z}[i]$$, the rings of Gaussian integers. If $$R$$ is any of these last two rings then the quotient group $$\text{SL}_2(R)/[\text{GL}_2(R), \text{GL}_2(R)] = \text{E}_2(R)/[\text{GL}_2(R), \text{GL}_2(R)]$$ has two elements, which is Hua and Reiner's result [1] when specializing $$R$$ to $$\mathbb{Z}$$.

• [1] L. Hua and I. Reiner, "Automorphisms of the unimodular group", 1951.
• [2] P. M. Cohn, "On the structure of the $$GL_2$$ of a ring", 1966.
• [3] R. Heitmann, "Generating non-Noetherian modules efficiently", 1984.
• [4] B. Magurn, "An algebraic introduction to $$K$$-theory", 2002.