Two-sided properties of elements in exchange rings. (Q495874)

The authors prove a theorem that describes the behaviour of elements of an exchange ring. In particular, the theorem they prove leads to the following results. (1.) If \(R\) is an exchange ring and \(a\in R\), then there exists an idempotent \(e\in aRa\) such that \(1-e\in (1-a)R(1-a)\). (2.) A ring \(R\) is an exchange ring if and only if, for every \(a\in R\), there exists an idempotent \(e\in Ra\) such that \(1-e\in (1-a)R\). Lastly, these results are used to study the endomorphism rings of exchange modules.