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.