Rings with quasi-projective left ideals

Abstract

A ring R is a left qp -ring if each of its left ideals is quasi-projective as a left R -module in the sense of Wu and Jans. The following results giving the structure of left qp -rings are obtained. Throughout R is a perfect ring with radical N: (1) Let R be local. Then R is a left qp -ring iff N2 = (0) or R is a principal left ideal ring with dec on left ideals, (2) If R is a left qp -ring and T is the sum of all those indecomposable left ideals of R which are not projective, then T is an ideal of R and N = ΓφL, L is a left ideal of R such that every left subideal of L is projective, R/T is hereditary, and R is heredity iff T = (0). (3) If R is left qp -ring then R = (? *£), where S is hereditary, T is a direct sum of finitely many local qp -rings and M is a (S, Γ)-bimodule. (4) A perfect left qp-ring is semi-primary. (5) Let R be an indecomposable ring such that it admits a faithful projective injective left module. Then R is a left qp-ring iff R is a local principal left ideal ring or R is a left-hereditary ring with dec on left ideals. (6) Let R be an indecomposable QF-ring. Then R is a left qp-ring if each homomorphic image of R is a g-ring (each one-sided ideal is quasi-injective). (7) If a left ideal A of left spring R is not projective then the projective dimension of A is infinite, thus Igi dim/? =0,1, or °c. An example of a left artinian left qp-ring which is not right qp-ring is also given.