Characterizing categoricity in the class $Add(M)$

We show that the condition of being categorical in a tail of cardinals can be characterized for the class of $R$-modules of the form $\Add(M)$. More precisely, let $R$ be a ring and $M$ be an $R$-module which can be generated by $\leq \aleph$ elements. Then $\Add(M)$ is $κ$-categorical in all $κ>\Vert R\Vert+\aleph+\aleph_0$ if and only if $\Add(M)$ is $κ$-categorical in some $κ>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $κ$ in $\Add(M)$ is $M$-free for all $κ>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $(\Vert R\Vert+\aleph+\aleph_0)^{+}$ in $\Add(M)$ is $M$-free. As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if the module $P^{(\aleph_0)}$ is free for each countably generated pure-projective $R$-module $P$; the class of semisimple $R$-modules is categorical in some (all) big cardinal if and only if $R$ admits a unique simple module up to isomorphism, partly answering a question proposed in [5, Mazari-Armida M., Characterizing categoricity in several classes of modules. J. Algebra 617, 382-401 (2023)].