We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is -free if every subset of size 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 -free abelian groups G. This result includes as a special case the existence of non-free separable -free abelian groups G (e.g. with End G = ℤ ⊕ Fin G), known until recently only for k = 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-1-3,
author = {Daniel Herden and H\'ector Gabriel Salazar Pedroza},
title = {$\_k$-free separable groups with prescribed endomorphism ring},
journal = {Fundamenta Mathematicae},
volume = {228},
year = {2015},
pages = {39-55},
zbl = {06451631},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-1-3}
}
Daniel Herden; Héctor Gabriel Salazar Pedroza. $ℵ_k$-free separable groups with prescribed endomorphism ring. Fundamenta Mathematicae, Tome 228 (2015) pp. 39-55. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-1-3/