Rings with internal cancellation

In this paper, we study the class of rings that satisfy internal direct sum cancellation with respect to their 1-sided ideals. These are known to be precisely the rings in which regular elements are unit-regular. Further characterizations for such “IC rings” are given, in terms of suitable versions of stable range conditions, and unique generator properties of idempotent generated right ideals. This approach leads to a uniform treatment of many of the known characterizations for an exchange ring to have stable range 1. Rings whose matrix rings are IC turn out to be precisely those rings whose finitely generated projective modules satisfy cancellation. We also offer a couple of “hidden” characterizations of unit-regular elements in rings that shed some new light on the relation between similarity and pseudo-similarity—in monoids as well as in rings. The paper concludes with a treatment of ideals for which idempotents lift modulo all 1-sided subideals. An appendix by R.G. Swan1on the failure of cancellation for finitely generated projective modules over complex group algebras shows that such algebras are in general not IC.