Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings.

One direction would be the following. Consider $\mathbb{N}$ (with the French convention, i.e. including $0$), and let $n\in\mathbb{N}$.
Consider all algorithms based on the following operations:

* adding $1$,
* multiplying two variables,
* summing two variables

and allowed to have any number of local variables (no loops or other advanced programming, just allocation of variables -for free- and the above three operations). Let $\delta(n)$ be the least number of operations needed for such an algorithm to return $n$. 

>What can we say about the function $\delta$? Can it be generalized to other rings or semi-rings? Can it be turned into a distance? Does it say something about the algebra of the given ring or semi-ring? 

It feels a little bit like Kolmogorov complexity (except that I chose not to allow taking differences of variables, which is questionable).


There could be many other ways to turn finitely generated rings or semi-rings into geometric object, I would be interested in any of them.