  Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is
called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$
with flat dimension at most $n$. In this paper, we prove first that
($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory,
where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all
$R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective
$R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and
$L_n$-global dimension of a ring. Finally, over rings with weak global
dimension $\leq n$, perfect rings, and $L_n$-hereditary rings, more properties
and applications of $L_{n}$-injective modules, $L_{n}$-injective dimensions of
modules and $\mathcal{L}_{n}$-global dimensions of rings are given.
