On the standard form for matrices of order two

We establish a criterion for a subring of the field of rational numbers to have a unique standard form (in the sense of Cohn). A similar criterion is obtained for quotient rings of the ring of integers.
Definition 1. Let $R$ be an associative ring with unit, $C \in GL_2(R)$ and
$$ C=\begin{pmatrix}\alpha& 0\\ 0&\beta\end{pmatrix}\begin{pmatrix}a_1& 1\\ -1& 0\end{pmatrix}\begin{pmatrix}a_2& 1\\ -1& 0\end{pmatrix}\dots\begin{pmatrix}a_t& 1\\ -1& 0\end{pmatrix}, $$
where $t \geqslant 0$. Suppose that the following conditions are satisfied:
1) $\alpha$ and $\beta$ are invertible in $R$;
2) if $1 < i < t$, then $a_i$ is a nonzero non-invertible element of $R$;
3) if $t = 2$, then $a_1$ and $a_2$ cannot both be $0$.
Then the above representation is said to be a standard form for $C$.
Definition 2. 1) A ring $R$ is said to have a unique standard form if no matrix $C \in GL_2(R)$ can be represented by two different standard forms.
2) A ring $R$ is said to be quasi-free if the identity matrix $E \in GL_2(R)$ does not possess a nontrivial standard form.
Theorem 5. If a ring $R$ is quasi-free, then for every nonzero non-invertible elements $b$ and $c$ of $R$ the element $bc-1$ is non-invertible in $R$.
Theorem 5 enables us to prove Proposition 7 and Theorem 8.
Proposition 7. Let $R =\mathbf{Z}/n\mathbf{Z}$, where $n > 1$. The following conditions are equivalent:
a) $R$ has a unique standard form;
b) $R$ is quasi-free;
c) $n$ is a prime.
Theorem 8. 1) A subring of the field $\mathbf{Q}$ is quasi-free if and only if it coincides with $\mathbf{Q}$ or with $\mathbf{Z}$. 2) A subring of the field $\mathbf{Q}$ has a unique standard form if and only if it coincides with $\mathbf{Q}$.