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

Rüdiger Göbel¹ ; Daniel Herden² ; Héctor Gabriel Salazar Pedroza³

¹ ($\dagger$July 28, 2014)
² Department of Mathematics Baylor University One Bear Place #97328 Waco, TX 76798-7328, U.S.A.
³ Mathematical Institute Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland

Fundamenta Mathematicae, Tome 231 (2015) no. 1, pp. 39-55.

Résumé

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 $k=1$.

Détail

DOI : 10.4064/fm231-1-3

Keywords: consider unital rings additive group want construct zfc each natural number family aleph k free a modules which separable abelian groups special decompositions recall a module aleph k free every subset size aleph contained submodule refine definition separable abelian group finite subset contained direct summand despite module almost admits many decompositions able control endomorphism ring mathop end its additive structure strong able arbitrarily large mathop end oplus mathop fin mathop end mathop fin where mathop fin ideal mathop end endomorphisms finite rank special choice permits interesting separable aleph k free abelian groups result includes special existence non free separable aleph k free abelian groups mathop end mathbb oplus mathop fin known until recently only