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

Abstract

top
We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of k -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is k -free if every subset of size < 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 k -free abelian groups G. This result includes as a special case the existence of non-free separable k -free abelian groups G (e.g. with End G = ℤ ⊕ Fin G), known until recently only for k = 1.

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.