Characterizing categoricity in several classes of modules

DOI10.1016/J.JALGEBRA.2022.10.031arXiv2202.07900MaRDI QIDQ6391264

Marcos Mazari-Armida

Publication date: 16 February 2022

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 $all$ $lambda>|R|+alep{h}_0$ if and only if $Rcong{M}_n(D)$ for $D$ a division ring and $ngeq1$. 2. The class of flat $R$-modules is $lambda$-categorical in $all$ $lambda>|R|+alep{h}_0$ if and only if $Rcong{M}_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $ngeq1$. 3. Assume $R$ is a commutative ring. The class of absolutely pure $R$-modules is $lambda$-categorical in $all$ $lambda>|R|+alep{h}_0$ 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.

Mathematics Subject Classification ID

Model-theoretic algebra (03C60) Noncommutative local and semilocal rings, perfect rings (16L30) Applications of logic to commutative algebra (13L05) Classification theory, stability, and related concepts in model theory (03C45) Noetherian rings and modules (associative rings and algebras) (16P40) Applications of logic in associative algebras (16B70) Categoricity and completeness of theories (03C35)

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