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. It is obtained from the lattice of the principal ideals of A. If such 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-arguesian projective plane.
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 $1$).