On prime spectrums of commutative rings

摘要

We show that pi-regular rings and clean rings can be completely characterized by topological properties of their prime spectrums respectively. In addition, we give some applications of those result. Among others, we improve the main result of Samei (2004) and give a new criterion for a clean ring that a commutative ring is clean if and only if idempotents lifts modulo every radical ideal.