2019/03/01 by Kucera, Thomas G., Mazari-Armida, Marcos
#03C45 #03C48 #03C60 #13L05 #16D10 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1903.00414
We show that certain classes of modules have universal models with respect to pure embeddings. Theorem. Let R be a ring, T a first-order theory with an infinite model extending the theory of R-modules and KT=(Mod(T), ≤pp) (where ≤pp stands for pure submodule). Assume KT has joint embedding and amalgamation. If λ|T|=λ or ∀ μ< λ( μ|T| < λ), then KT has a universal model of cardinality λ. As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of R-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer Question 4.25 of [Maz]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.