Comparability, separativity, and exchange rings.

There are several long-standing open problems which ask whether regular rings, and C*-algebras of real rank zero, satisfy certain module cancellation properties. Ara, Goodearl, O'Meara and Pardo recently observed that both types of rings are exchange rings, and showed that separative exchange rings have these good cancellation properties thus answering the questions affirmatively in the separative case. In this article, we prove that, for any positive integer s, exchange rings satisfying s-comparability are separative, thus answering the questions affirmatively in the s-comparable case. We also introduce the weaker, more technical, notion of generalized s-comparability, and show that this condition still implies separativity for exchange rings. On restricting to directly finite regular rings, we recover results of Ara, O'Meara and Tyukavkin.