Characterizing categoricity in several classes of modules

TL;DR

This paper characterizes when certain classes of modules are categorical in large cardinals, linking algebraic properties of rings to model-theoretic categoricity, and confirms Shelah's Conjecture for these classes.

Contribution

It provides algebraic characterizations of categoricity in modules for several classes, extending Shelah's Conjecture beyond first-order axiomatizable classes.

Findings

Locally pure-injective modules are categorical iff the ring is a matrix algebra over a division ring.

Flat modules are categorical iff the ring is a matrix algebra over a local ring with a nilpotent maximal ideal.

Absolutely pure modules are categorical iff the ring is a local artinian ring.

Abstract

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective $R$-modules is $\lambda$-categorical in $all$ $\lambda > |R|+\aleph_0$ if and only if $R \cong M_n(D)$ for $D$ a division ring and $n \geq 1$. 2. The class of flat $R$-modules is $\lambda$-categorical in $all$ $\lambda > |R| + \aleph_0$ if and only if $R \cong M_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $n \geq 1$. 3. Assume $R$ is a commutative ring. The class of absolutely pure $R$-modules is $\lambda$-categorical in $all$ $\lambda > |R| + \aleph_0$ if and only if $R$ is a local artinian ring. We show that in the above results it is enough to assume $\lambda$-categoricity in $some$ large cardinal $\lambda$. This shows that Shelah's Categoricity Conjecture holds for the class of locally pure-injective modules, flat modules and absolutely pure modules. These classes are not first-order axiomatizable for arbitrary rings. We provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable.