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This answer provides references for the facts mentioned in the comments. 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.

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.

This answer provides references for the facts mentioned in the comments. 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., $R = \mathbb{Z}[\frac{1}{2n}]$ where $n \in \mathbb{N}_{> 0}$ or $R = k[X]$ where $k$ is a field with at least $3$ elements). See also , among others, Cohn's generalization [2, Theorem 9.4] of the result of Hua and Reiner [1].

• [1] L. Hua and I. Reiner, "Autommorphisms 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.

This answer provides references for the facts mentioned in the comments. 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.

This answer provides references for the facts mentioned in the comments. More on this topic can be found in [1, 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 [1, 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 $d + 1$ [2, Corollary 2.3], the result follows from the previous claim.

For $n = 2$, Section 9 of P. M. Cohn's [3] contains a wealth of valuable results. For instance it follows from [3, 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., $R = \mathbb{Z}[\frac{1}{2n}]$ where $n \in \mathbb{N}_{> 0}$ or $R = k[X]$ where $k$ is a field with at least $3$ elements). See also , among others, Cohn's generalization of the result of Hua and Reiner [1, Theorem 9.4].

• [1] P. M. Cohn, "On the structure of the $GL_2$ of a ring", 1966.
• [2] R. Heitmann, "Generating non-Noetherian modules efficiently", 1984.
• [3] B. Magurn, "An algebraic introduction to $K$-theory", 2002.

This answer provides references for the facts mentioned in the comments. 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., $R = \mathbb{Z}[\frac{1}{2n}]$ where $n \in \mathbb{N}_{> 0}$ or $R = k[X]$ where $k$ is a field with at least $3$ elements). See also , among others, Cohn's generalization [2, Theorem 9.4] of the result of Hua and Reiner [1].

• [1] L. Hua and I. Reiner, "Autommorphisms 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.