2022/02/16 by Mazari-Armida, Marcos · 1 citation
#03C35 #03C45 #03C60. Secondary: 16L30 #13L05 #16P40 #FOS: Mathematics #Logic (math.LO) #Primary: 16B70 #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2202.07900
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 λ-categorical in all λ> |R|+ℵ0 if and only if R ≅ Mn(D) for D a division ring and n ≥ 1. 2. The class of flat R-modules is λ-categorical in all λ> |R| + ℵ0 if and only if R ≅ 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 λ-categorical in all λ> |R| + ℵ0 if 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 large 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.