Thanks for your reply Todd. Do I need to edit my post ? I tried to find counter exemple using your remark, but failed. Do you know if there is a method to compute the number of sub vector space in a finite vs ?

Oh ! Ok, you're right, thanks a lot :) I had a look first in the book of Bruce Blackadar. He took the Murray Von Naumann equivalence to define $V(A)$. However those two semi groups lead to the same group and to the same K theory.

but $V(A)$ is always torsion free right ? The only projection equivalent to $0$ is $0$ itself.