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

Daniel Herden, and Héctor Gabriel Salazar Pedroza. "$ℵ_k$-free separable groups with prescribed endomorphism ring." Fundamenta Mathematicae 231.1 (2015): 39-55. <http://eudml.org/doc/282892>.

@article{DanielHerden2015,
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 $ℵ_k$-free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is $ℵ_k$-free if every subset of size $< ℵ_k$ 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 End G of its additive structure in a strong way: we are able to find arbitrarily large G with End G = A ⊕ Fin G (so End G/Fin G = A, where Fin G is the ideal of End G of all endomorphisms of finite rank) and a special choice of A permits interesting separable $ℵ_k$-free abelian groups G. This result includes as a special case the existence of non-free separable $ℵ_k$-free abelian groups G (e.g. with End G = ℤ ⊕ Fin G), known until recently only for k = 1.},
author = {Daniel Herden, Héctor Gabriel Salazar Pedroza},
journal = {Fundamenta Mathematicae},
keywords = {prediction principles; almost free abelian groups; endomorphism rings},
language = {eng},
number = {1},
pages = {39-55},
title = {$ℵ_k$-free separable groups with prescribed endomorphism ring},
url = {http://eudml.org/doc/282892},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Daniel Herden
AU - Héctor Gabriel Salazar Pedroza
TI - $ℵ_k$-free separable groups with prescribed endomorphism ring
JO - Fundamenta Mathematicae
PY - 2015
VL - 231
IS - 1
SP - 39
EP - 55
AB - We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of $ℵ_k$-free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is $ℵ_k$-free if every subset of size $< ℵ_k$ 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 End G of its additive structure in a strong way: we are able to find arbitrarily large G with End G = A ⊕ Fin G (so End G/Fin G = A, where Fin G is the ideal of End G of all endomorphisms of finite rank) and a special choice of A permits interesting separable $ℵ_k$-free abelian groups G. This result includes as a special case the existence of non-free separable $ℵ_k$-free abelian groups G (e.g. with End G = ℤ ⊕ Fin G), known until recently only for k = 1.
LA - eng
KW - prediction principles; almost free abelian groups; endomorphism rings
UR - http://eudml.org/doc/282892
ER -