Characterizing categoricity in the class $Add(M)$

TL;DR

This paper characterizes when classes of modules, specifically additive modules over a ring, are categorical in large cardinals, linking categoricity to module freeness and providing applications to pure-projective and semisimple modules.

Contribution

It provides a precise characterization of categoricity in tail cardinals for classes of additive modules, connecting it to module freeness and answering open questions.

Findings

Categoricity in large cardinals is equivalent to all modules being free in the class.

Pure-projective modules are categorical iff certain free modules exist.

Semisimple modules are categorical iff the ring has a unique simple module.

Abstract

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 $\kappa$-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 $M$-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 $M$-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)].