Comparability, separativity, and exchange rings

Elements in Exchange Rings with Related Comparability

We show that if R is an exchange ring, then the following are equivalent: (1) R satisfies related comparability. (2) Given a,b,d ∈ R with aR+bR = dR, there exists a related unit w ∈ R such that a+bt = dw. (3) Given a,b ∈ R with aR = bR, there exists a related unit w ∈ R such that a= bw. Moreover, we investigate the dual problems for rings which are quasi-injective as right modules.

Countable Exchange and Full Exchange Rings

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an א0-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring 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 excha...

Exchange rings, exchange equations, and lifting properties

Separative Ideals of Exchange Rings

separative ideals of exchange rings