# ${\aleph}_{k}$-free separable groups with prescribed endomorphism ring

Daniel Herden; Héctor Gabriel Salazar Pedroza

Fundamenta Mathematicae (2015)

- Volume: 231, Issue: 1, page 39-55
- ISSN: 0016-2736

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topDaniel 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 -

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