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
dcc 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 = T ⊕ L, 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,T)-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 dcc
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 q-ring (each one-sided ideal
is quasi-injective). (7) If a left ideal A of left qp-ring R is not projective
then the projective dimension of A is infinite, thus lgl.dimR = 0,1, or ∞.
An example of a left artinian left qp-ring which is not right qp-ring is also
given.
, where 