Exchange rings and their extensions
An exchange ring is a ring \(R\) for which the module \(R_R\) (equivalently, \(_RR\)) has the finite exchange property [cf. \textit{R. B. Warfield}, jun., Math. Ann. 199, 31-36 (1972; Zbl 0228.16012)]. The authors first study the (von Neumann) regularity of such a ring. Their main theorem states that if \(R\) is an exchange ring whose primitive factor rings are all Artinian, and if all homomorphic images of \(R\) are semiprimitive, then \(R\) is regular -- in fact, unit-regular. This extends a result of \textit{H.-P. Yu} [Commun. Algebra 25, No. 2, 661-670 (1997; Zbl 0873.16007)], who had proved that under the above hypotheses, \(R\) is strongly \(\pi\)-regular. Applications concerning strong regularity and strong \(\pi\)-regularity follow from the main theorem. The second section of the paper is concerned with transferring the exchange property from a ring \(R\) to related rings. For example, the authors prove that if \(\alpha\) is a ring endomorphism of \(R\), then \(R\) is an exchange ring if and only if the skew power series ring \(Rx;\alpha\) is an exchange ring.
scientific article; zbMATH DE number 1895985