There is the general notion of a rank ring.
This was introduced by von Neumann in the study of continuous geometry. In short, a continuous geometry is an irreducible complemented complete modular lattice. By a theorem from von Neumann, it is obtained from the lattice of the right principal ideals of some von Neumann regular ring $A$. (If such a lattice is simple of finite length, then it is the lattice of subspaces of a vector space $F^n$, $F$ being a not necessarily commutative field, or the lattice associated to a non-Desarguesian projective plane.) Moreover, the ring $A$ is a rank ring.
The book of von Neumann, Continuous geometry, may be a starting point. There are many structural results for matrix rings over noncommutative rings (themselves matrix rings of matrix rings of matrix rings of ...) leading to a space with "linear" subspaces of any real dimension between $0$ and $1$.
