On PP-rings

A ringR is called a leftPP-ring if every principal left ideal is projective, equivalently if the left annihilatorl(a) is generated by an idempotent for allaR. These rings seem first to have been discussed by Hattori 2] and examples include (von Neumann) regular rings and domains (possibly noncommutative). In this note we give a new characterization of leftPP-rings, use that to give an elementary proof of a result of Xue 4] characterizing triangularPP-rings, and then determine when the ringT _n (R) of upper triangular matrices overR is a leftPP-ring. Throughout the paper all rings have a unity and all modules are unitary.This research was supported by NSERC Grant A8075