  Given an additive category $\mathcal{C}$ and an integer $n\geqslant 2$. We
form a new additive category $\mathcal{C}[\epsilon]^n$ consisting of objects
$X$ in $\mathcal{C}$ equipped with an endomorphism $\epsilon_X$ satisfying
${\epsilon^n_X}=0$. First, using the descriptions of projective and injective
objects in $\mathcal{C}[\epsilon]^n$, we not only establish a connection
between Gorenstein flat modules over a ring $R$ and $R[t]/(t^n)$, but also
prove that an Artinian algebra $R$ satisfies some homological conjectures if
and only if so does $R[t]/(t^n)$. Then we show that the corresponding homotopy
category $\K(\mathcal{C}[\epsilon]^n)$ is a triangulated category when
$\mathcal{C}$ is an idempotent complete exact category. Moreover, under some
conditions for an abelian category $\mathcal{A}$, the natural quotient functor
$Q$ from $\K(\mathcal{A}[\epsilon]^n)$ to the derived category
$\D(\mathcal{A}[\epsilon]^n)$ produces a recollement of triangulated
categories. Finally, we prove that if $\mathcal{A}$ is an Ab4-category with a
compact projective generator, then $\D(\mathcal{A}[\epsilon]^n)$ is a compactly
generated triangulated category.
