Characterizing categoricity in the class $Add(M)$

Abstract

We study categoricity of the additive closure $\operatorname{Add}(M)$, consisting of all direct summands of arbitrary direct sums of copies of a fixed module $M$. For an $\eta$-generated module $M$, we prove that categoricity in a single cardinal $\lambda \ge \theta = \max\{|R|,\eta\}$ is equivalent to the stabilization condition $P^{(\eta)} \cong M^{(\eta)}$ for every nonzero $\eta$-generated $P \in \operatorname{Add}(M)$, or equivalently $\operatorname{Add}(P)=\operatorname{Add}(M)$. This condition implies that every $X \in \operatorname{Add}(M)$ of cardinality $\lambda > \nu_M = \max\{\theta, |\mathcal{S}_\eta(M)|\}$ is isomorphic to $M^{(\lambda)}$, where $\mathcal{S}_\eta(M)$ is a representative set of the nonzero $\eta$-generated modules in $\operatorname{Add}(M)$. Since $|\mathcal{S}_\eta(M)| \le 2^\theta$, we obtain a uniform ZFC tail above $2^\theta$. For projective modules, the criterion reduces to Bass's right $p$-connectedness. For pure-projective modules, it forces von Neumann regularity and simplicity: the pure-projective modules are eventually categorical if and only if $R$ is a simple von Neumann regular ring. In the commutative case, this recovers precisely the fields. Our approach combines Walker's decomposition theorem, support reduction, and Eilenberg absorption, and we highlight several subtle pitfalls in naive downward transfer arguments.