I'm not sure that this is the answer you are looking for, but you might be interested in Mnev's universality theorem. Unwinding the language of semialgebraic sets, this says the following: Consider any finite set of polynomial equations and inequalities with integer coefficients, such as $$\zeta^3=1 \quad \zeta \neq 1.$$

Then there is an arrangement $A$ of points and lines such that $A$ can be realized over a commutative ring $R$ if and only if $R$ contains a solution to those equalities and inequalities.

For example, the Fano plane is representable only if $2=0$ in $R$. The statement "you can find $9$ points $x(i,j)$ labeled by $\mathbb{F}_3^2$ such that $x(i_1, j_1)$, $x(i_2, j_2)$ and $x(i_3, j_3)$ are collinear if and only if $\sum (i_s, j_s)=0$" is equivalent to the statement "$R$ contains a nontrivial cube root of unity."

But all of this is for $R$ commutative; I don't know if there is a noncommutative version of Mnev.

I'm not sure that this is the answer you are looking for, but you might be interested in Mnev's universality theorem. Unwinding the language of semialgebraic sets, this says the following: Consider any finite set of polynomial equations and inequalities with integer coefficients, such as $$\zeta^3=1 \quad \zeta \neq 1.$$

Then there is an arrangement $A$ of points and lines such that $A$ can be realized over a commutative ring $R$ if and only if $R$ contains a solution to those equalities and inequalities.

For example, the Fano plane is representable only if $2=0$ in $R$. The statement "you can find $9$ points $x(i,j)$ labeled by $\mathbb{F}_3^2$ such that $x(i_1, j_1)$, $x(i_2, j_2)$ and $x(i_3, j_3)$ are collinear if and only if $\sum (i_s, j_s)=0$" is equivalent to the statement "$R$ contains a nontrivial cube root of unity."

But all of this is for $R$ commutative; I don't know if there is a noncommutative version of Mnev.