A ring $R$ is said to be left $n$-semihereditary, if every
$n$-generated left ideal of $R$ is projective. It is shown that
for a ring $R$, the following statements are equivalent: (1) $R$
is left $n$-semihereditary; (2)every $n$-generated submodule of a
 projective left $R$-module is projective; (3)every torsion-less right
$R$-module is $n$-flat; (4)$R$ is left $n$-coherent and every
$n$-generated right ideal of $R$ is flat; (5)$R$ is left
$n$-coherent and every right ideal of $R$ is $n$-flat; (6)every
factor module of an $n$-injective left $R$-module is
$n$-injective; (7)the sum of an arbitrary family of $n$-injective
submodules of a left $R$-module is $n$-injective. Moreover, some
new characterizations of Pr$\rm\ddot{u}$fer rings are given.