As an alternative to YCor's beautiful answer, one can use the following theorem of P. M Cohn [1, Theorem 9.5].
Theorem. Let $R$ be a ring which is quasi-free for $\text{GE}_2$ and denote by $N$ the ideal generated by all $\alpha - 1$ with $\alpha \in U(R)$. Then there is a split exact sequence $$0 \rightarrow R/N \rightarrow \text{GE}_2(R)^\text{ab} \rightarrow U(R)^\text{ab} \rightarrow 0.$$ and the mapping $\alpha \mapsto \begin{pmatrix} \alpha^\text{ab} & 0 \\ 0 & 1 \end{pmatrix}$ induces a splitting.
In the above statement
- $U(R)$ denotes the unit group of $R$,
- $G^\text{ab}$ denotes the abelianization of a group $G$ and $g \mapsto g^\text{ab}$ the abelianization homomorphism (in particular in the case of $U(R)\to U(R)^\text{ab}$),
- $\text{GE}_2(R)$ is the subgroup of $\text{GL}_2(R)$ generated by the elementary matrices $\begin{pmatrix} 1 & r \\ 0 & 1 \end{pmatrix}$, $\begin{pmatrix} 1 & 0 \\ s & 1 \end{pmatrix}$ with $r,s \in R$ and the diagonal matrices $\begin{pmatrix} \alpha & 0 \\ 0 & \beta \end{pmatrix}$ with $\alpha, \beta \in U(R)$.
I will not expand the definition of a quasi-free ring for $\text{GE}_2$ but just mention that a discretely normed ring is a quasi-free for $\text{GE}_2$. In particular $K[X]$ is quasi-free for $\text{GE}_2$ if $K$ is a field.
Note that if $\text{SL}_2(R) = \text{E}_2(R)$$\text{SL}_2(R)$ is generated by the elementary matrices, e.g. $R$ is Euclidean, then $\text{GE}_2(R) = \text{GL}_2(R)$. This holds in particular for $R = K[X]$ with $K$ a field since $K[X]$ is Euclidean (only) in this case.
If $K = \mathbf{F}_2$,$K$ is the field with two elements, then the above theorem yields an isomorphism of the additive group of $K[X]$ with $\text{GL}_2(K[X])^\text{ab}$. If $K$ is a field with more than $2$two elements, the same theorem yields an isomorphism of $\text{GL}_2(K[X])^\text{ab}$ with $U(K)$.
[1] P. M. Cohn, "On the structure of the $\mathrm{GL}_2$ of a ring", 1966 (MSN).
