A model theoretic solution to a problem of L\'{a}szl\'{o} Fuchs

DOI10.1016/J.JALGEBRA.2020.09.029arXiv2005.07120MaRDI QIDQ6340722

Author name not available (Why is that?)

Publication date: 14 May 2020

Abstract: Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $lambda$ such that there is a universal abelian $p$-group for purity of cardinality $lambda$, i.e., an abelian $p$-group $U_{l}ambda$ of cardinality $lambda$ such that every abelian $p$-group of cardinality $leqlambda$ purely embeds in $U_{l}ambda$. In this paper we use ideas from the theory of abstract elementary classes to show: $extbfTheorem.$ Let $p$ be a prime number. If $lambd{a}^{alep{h}_0}=lambda$ or $forallmu<lambda(m{u}^{alep{h}_0}<lambda)$, then there is a universal abelian $p$-group for purity of cardinality $lambda$. Moreover for $ngeq2$, there is a universal abelian $p$-group for purity of cardinality $alep{h}_n$ if and only if $2^{alep{h}_0}leqalep{h}_n$. As the theory of abstract elementary classes has barely been used to tackle algebraic questions, an effort was made to introduce this theory from an algebraic perspective.

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