Simple (von Neumann) regular rings (with 1) are discussed, from the point of view of rank and dimension functions. In particular, conditions under which the ring possesses a unique rank function are investigated. Sub- and over-additive functions resembling rank functions are introduced. A short proof of Halperin's result that a matrix ring over a rank ring is a rank ring in a natural manner is presented. Two special examples of rank-like functions, emanating from the isomorphism classes of the finitely generated protective modules are examined, and it is shown that a simple unit-regular ring admits a metrizable topological ring structure, induced by the isomorphism classes of its projective modules. There is also a natural over-additive function obtained from these classes, and when this function defines a uniform topology on the ring (which does not always occur), the ring possesses a rank function; this is proved via the Alexandroff-Urysohn metrization theorem. As a consequence, we obtain the following results. (1) If there exists an integer k such that for all principal right ideals J, K of the simple unit-regular ring R, either J is embeddable (as a module) in k copies of K or the same with J, K switched, then R possesses a unique rank function. (2) The over-additive function described above is a rank function on the simple unit-regular ring R if and only if all matrix rings over R satisfy: For all rational numbers p greater than 1, for all principal right ideals J, K, there exists an integer m such that mp is an integer and either m copies of J are embeddable in mp copies of K, or the same with J and K reversed. This is a weakened comparability condition.