  In this paper, we introduced a generalization of the derived category, which
is called the $n$-derived category and denoted by $\D_{n}(R)$, of a given ring
$R$ for each $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring
is proved to be very closely connected with its left little finitistic
dimension. We also introduce and investigate the notions of $n$-exact
sequences, $n$-projective (resp., $n$-injective) modules and $n$-exact
complexes. In particular, we characterize the left little finitistic dimensions
in terms of all above notions. Finally, we build a connection of the classical
derived categories and $n$-derived categories.
