$ℵ_k$-free separable groups with prescribed endomorphism ring

• Autorzy: Daniel Herden , Héctor Gabriel Salazar Pedroza
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 16Dxx: Modules, bimodules and ideals
• 03E75: Applications of set theory
• 20Kxx: Abelian groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2015
• Tom: 231
• Numer: 1
• Strony: 39-55

Daty

wydano

2015

Twórcy

autor

Daniel Herden

• Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, U.S.A.

autor

Héctor Gabriel Salazar Pedroza

• Mathematical Institute, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

Identyfikatory

DOI

10.4064/fm231-1-3