CCR-rings (Q1121356)

A ring is called \(\pi\)-regular (strongly regular, resp.) if for every a there exists an element x and an integer n such that \(a^ nxa^ n=a^ n\) \((a^ 2x=a\), resp.). It is CCR if every primitive homomorphic image is simple with a minimal one-sided ideal. The main result claims that in a \(\pi\)-regular semisimple CCR-ring R there exists a nonzero idempotent e such that eRe is strongly regular. It is noticed that a ring is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements.