On the problem of stability of abstract elementary classes of modules

It is an open problem of Mazari-Armida whether every abstract elementary class of $R$-modules $(\mathbf{K},{\le }_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}})$, with ${\le }_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}}$ the pure submodule relation, is stable. We answer this question in the negative by constructing unstable abstract elementary classes $(\mathbf{K},{\le }_{\mathrm{p}\mathrm{u}\mathrm{r}\mathrm{e}})$ of torsion-free abelian groups. On the other hand, we prove (in $\mathrm{Z}\mathrm{F}\mathrm{C}$) that if $R$ is any ring and $(\mathbf{K},\preccurlyeq )$ is an abstract elementary class of $R$-modules which is $\kappa$-local (also called $\kappa$-tame) for some $\kappa \ge \mathrm{L}\mathrm{S}(\mathbf{K},\preccurlyeq )$, then $(\mathbf{K},\preccurlyeq )$ is almost stable, where almost stability is a new notion of independent interest that we introduce in this paper, and which is equivalent to the usual notion of stability under the assumption of amalgamation. As a consequence, assuming the existence of a strongly compact cardinal $\kappa$, we have that every abstract elementary class $(\mathbf{K},\preccurlyeq )$ of $R$-modules with amalgamation satisfying $\kappa >\mathrm{L}\mathrm{S}(\mathbf{K},\preccurlyeq )$ is stable.