  Let $R$ be a commutative noetherian ring and $\mathfrak{a}$ an ideal of $R$.
The goal of this paper is to establish the local-global principle for the
artinianness dimension $r_{\mathfrak{a}}(M)$, where $r_{\mathfrak{a}}(M)$ is
the smallest integer such that the local homology module of $M$ is not
artinian. For an artinian $R$-module $M$ with the set
$\mathrm{Coass}_{R}\mathrm{H}_{r_{\mathfrak{a}}(M)}^{\mathfrak{a}}(M)$ finite,
we show that
$r_{\mathfrak{a}}(M)=\mathrm{inf}\{r_{\mathfrak{a}R_{\mathfrak{p}}}(\mathrm{Hom}_{R}(R_{\mathfrak{p}},M))
\hspace{0.03cm}|\hspace{0.03cm}\mathfrak{p}\in \mathrm{Spec}R\}$. And the class
of all modules $N$ such that $\mathrm{Coass}_{R}N$ is finite is studied.
