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2015 | 231 | 1 | 39-55
Tytuł artykułu

$ℵ_k$-free separable groups with prescribed endomorphism ring

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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.
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  • Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, U.S.A.
  • Mathematical Institute, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-1-3
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