Characterizing categoricity in several classes of modules

Abstract

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. Theorem 0.1Assume R is an associative ring with unity.(1)The class of locally pure-injective R-modules is λ-categorical in all λ>card(R)+ℵ0if and only ifR≅Mn(D)for D a division ring andn≥1.(2)The class of flat R-modules is λ-categorical in all λ>card(R)+ℵ0if and only ifR≅Mn(k)for k a local ring such that its maximal ideal is left T-nilpotent andn≥1.(3)Assume R is a commutative ring. The class of absolutely pure R-modules is λ-categorical in all λ>card(R)+ℵ0if and only if R is a local artinian ring.We show that in the above results it is enough to assume λ-categoricity in some big cardinal λ. 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.