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? 

(the emphasis was added in edit, this is the part I am most interested in).

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

**Edit** Another direction would be to consider the minimal numbers $||n||$ of $1$s needed to write $n$ by a well-formed formula with the symbols $1$ $+$ $\times$ $($ and $)$; this happens to lead to open questions, see [this answer on MO][1].

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.

**Added in edit** To clarify where I'm headed to, both cases show that there are natural ways to construct functions that can play the role of the distance to $0$; but I am not aware of any distance functions (taking two arguments) of this sort. Is there any that has been considered?

 
  [1]: https://mathoverflow.net/a/103171/4961