Constructive Matrix and Orderable Groups

We study into the relationship between constructivizations of an associative commutative ring $K$ with unity and constructivizations of matrix groups $GL_n(K)$ (general), $SL_n(K)$ (special), and $UT_n(K)$ (unitriangular) over $K$. It is proved that for $n\geqslant3$, a corresponding group is constructible iff so is $K$. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group $UT_n(K)$ over an orderly constructible commutative associative ring $K$ is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.