Note on P.P. Rings: (A Supplement to Hattori’s Paper)

A ring R is called, according to 2, a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring. In this short note we shall give some additional remarks to A. Hattori 2. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in 2, which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.