$\aleph _k$-free separable groups with prescribed endomorphism ring
Volume 231 / 2015
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 $k=1$.
