The local-global principle for the artinianness dimensions

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.