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$\aleph _k$-free separable groups with prescribed endomorphism ring

Volume 231 / 2015

Fundamenta Mathematicae 231 (2015), 39-55 MSC: Primary 16Dxx, 20Kxx; Secondary 03E75. DOI: 10.4064/fm231-1-3

Abstract

We will consider unital rings $A$ with free additive group, and want to construct (in ZFC) for each natural number $k$ a family of $\aleph _k$-free $A$-modules $G$ which are separable as abelian groups with special decompositions. Recall that an $A$-module $G$ is $\aleph _k$-free if every subset of size $<\aleph _k$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of $G$ is contained in a free direct summand of $G$. Despite the fact that such a module $G$ is almost free and admits many decompositions, we are able to control the endomorphism ring $\mathop {\rm End} G$ of its additive structure in a strong way: we are able to find arbitrarily large $G$ with $\mathop {\rm End} G=A\oplus \mathop {\rm Fin} G$ (so $\mathop {\rm End} G /\mathop {\rm Fin} G=A$, where $\mathop {\rm Fin} G$ is the ideal of $\mathop {\rm End} G$ of all endomorphisms of finite rank) and a special choice of $A$ permits interesting separable $\aleph _k$-free abelian groups $G$. This result includes as a special case the existence of non-free separable $\aleph _k$-free abelian groups $G$ (e.g. with $\mathop {\rm End} G=\mathbb {Z} \oplus \mathop {\rm Fin} G$), known until recently only for $k=1$.

Authors

• Rüdiger Göbel($\dagger$July 28, 2014)
• Daniel HerdenDepartment of Mathematics
Baylor University
One Bear Place #97328
Waco, TX 76798-7328, U.S.A.
e-mail
• Héctor Gabriel Salazar PedrozaMathematical Institute