On the standard form for matrices of order two

M. N. Zonov ; E. A. Timoshenko

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 59 (2019), pp. 5-10

Résumé

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 $\mathbf{Q}$.

Détail

Mots-clés : matrix
Keywords: standard form, general linear group.