Countable and Full Exchange Rings

Abstract

We show that a suitable ring with a ``nice'' topology, in which convergent limits of units are units, is an \aleph_0-exchange ring. We generalize the argument to show that a semi-regular ring, R, with a ``nice'' topology, is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite exchange has countable exchange. Also, all modules with Dedekind-finite, semi-regular endomorphism rings are full exchange modules.

15 pages