Reference request. Something like a rank for matrices over non-commutative rings I'm looking for some function $P: M \rightarrow R$, where $M$ is matrix ring over some non-commutative ring that would behave "like" rank. In particular:


*

*$P(A+B) \leq P(A)+P(B)$

*$P(A\times B) \leq P(A)+P(B)$

*$P(\alpha I) = n$, for $\alpha \neq 0$  (maybe with additional restrictions, like $\alpha$ being invertable)


Any reference would be of a great help.
 A: 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 $1$.
A: A rank function on a ring (which often is, but need not be, vN regular) satisfies $N(rs) \leq N(r), N(s)$, instead of condition (2) (assuming the $\times$ symbol means multiplication). A convenient reference for rank functions on regular rings is Goodearl, Von Neumann regular rings.
For C*-algebras, Cuntz (Dimension functions on simple C$^*$-algebras, Math Ann 1978) developed what amounts to the same thing; there is a large literature on this type of dimension function, especially with a version of K$_0$, called K$_0^*$, an important invariant, out of which dimension functions arose. And, I almost forgot, this led to Blackadar & Handelman, Dimension functions and traces on C$^*$-algebras J  Functional Analysis  (1982).
A: For any non-zero $A\in M_{m.n}(R)$, the inner rank of $A$ is defined as the least positive integer $r$ such that there are matrices $P\in M_{m,r}(R)$, $Q\in M_{r,n}(R)$ satisfying $A=PQ$.
It's not difficult to see this definition coincides with the usual rank if $R=\mathbb{C}$. And when $R$ is some non-commutative ring, this notion also behaves like a ”rank“. You can look its many nice properties in Section 5.4 in Cohn's book: Free ideal rings and localization in general rings, Cambridge University Press, 2006.
I hope it can help you.
