Characterizing categoricity in several classes of modules

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. Theorem 0.1. Assume R is an associative ring with unity. (1) The class of locally pure-injective R -modules is A -categorical in all A > card(R) + aleph 0 if and only if R similar to= Mn(D) for D a division ring and n > 1. (2) The class of flat R -modules is A -categorical in all A > card(R) + aleph 0 if and only if R similar to= Mn(k) for k a local ring such that its maximal ideal is left T -nilpotent and n > 1. (3) Assume R is a commutative ring. The class of absolutely pure R -modules is A -categorical in all A > card(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 A- categoricity in some big cardinal A. 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. (c) 2022 Published by Elsevier Inc.