Comparability and s-Comparability of Modules over Exchange Rings

In this paper, we investigate the comparability structure over exchange rings. It is shown that the subdirect product of an exchange ring with stable range one and an exchange ring satisfying the comparability is also an exchange ring satisfying the comparability. This provides a new class of exchange rings satisfying the comparability. Furthermore, we investigate the s-comparability over exchange rings. This generalizes the corresponding results of Goodearl and Chen.