Characterizing categoricity in the class $Add(M)$

We show that the condition of being categorical in a tail of cardinals can be chacterized 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 $\kappa$-categorical in all $\kappa>\Vert R\Vert+\aleph+\aleph_0$ if and only if $Add(M)$ is $\kappa$-categorical in some $\kappa>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $\kappa$ in $Add(M)$ is free for all $\kappa>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $(\Vert R\Vert+\aleph+\aleph_0)^{+}$ in $Add(M)$ is free. As an application, we show that the class of semisimple (or pure-projective) $R$-modules is categorical in some (all) big cardinal if and only if $R\cong M_n(D)$ where $D$ is a division ring and $n\geq 1$, partly answering an question proposed in [5].