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.

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 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.

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.