Bounded rank of C∗-algebras

We introduce a concept of the bounded rank (with respect to a positive constant) for unital C∗-algebras as a modification of the usual real rank and present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given n and K>0 there exists a separable unital C∗-algebra ZnK such that every other separable unital C∗-algebra of bounded rank with respect to K at most n is a quotient of ZnK.