Indecomposable Almost Free Modules—The Local Case

Göbel, Rüdiger; Shelah, Saharon

doi:10.4153/cjm-1998-039-7

Cited by 27 publications

(7 citation statements)

References 25 publications

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“…The following example goes back to [2] -see also [19, 2.4]. It is based on a construction, pioneered in [18], of large ℵ 1 -free modules over a discrete valuation domain (DVD) that possess only trivial endomorphisms, see [19, 20.19]. The main point of the example is that it presents a module M such that the class L = lim − → add M is not closed under direct summands, and hence L is not closed under countable direct limits, cf.…”

Section: Closure Under Direct Limits and The Class Limmentioning

confidence: 99%

Closure properties of $\varinjlim\mathcal C$

Positselski,

Prihoda,

Trlifaj

2021

Preprint

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Let C be a class of modules and L = lim − → C the class of all direct limits of modules from C. The class L is well understood when C consists of finitely presented modules: L then enjoys various closure properties. Our first goal here is to study the closure properties of L in the general case when C ⊆ Mod-R is arbitrary. Then we concentrate on two important particular cases, when C = add M and C = Add M , for an arbitrary module M .In the first case, we prove that limand F S is the class of all flat right Smodules. In the second case, limwhere S is the endomorphism ring of M endowed with the finite topology, F S is the class of all right S-contramodules that are direct limits of direct systems of projective right S-contramodules, and F ⊙ S M is the contratensor product of the right S-contramodule F with the discrete left S-module M .For various classes of modules D, we show that if M ∈ D then limAdd M , but the equality for an arbitrary module M remains open. Finally, we deal with the case when M is an (infinitely generated) tilting module, and consider the problem of whether lim − → Add M = Add M where Add M is the class of all pure-epimorphic images of direct sums of copies of M . We prove that the equality holds, e.g., for all tilting modules over Dedekind domains.

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Section: Closure Under Direct Limits and The Class Limmentioning

confidence: 99%

Closure properties of $\varinjlim\mathcal C$

Positselski,

Prihoda,

Trlifaj

2021

Preprint

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“…In this section we explain the underlying geometry of our construction which was used also in [14], see there for further details.…”

Section: Topology Trees and A Forestmentioning

confidence: 99%

“…Following [14] we use the Definition 3.1 Let x ∈ B Λ be any element in the completion of the base algebra B Λ . Moreover, let η ∈ V α with α < λ.…”

Section: The Constructionmentioning

confidence: 99%

“…The construction of ℵ 1 -free E-rings R in ZFC is much easier if |R| = 2 ℵ 0 , because in case |R| = ℵ 1 we are closer to freeness, a property which tries to prevent endomorphisms from being scalar multiplication. Thus we need more algebraic arguments and will utilize a combinatorial prediction principle similar to the one used by the first two authors in [14] for constructing almost-free groups of cardinality ℵ 1 with prescribed endomorphism rings.…”

Section: Introductionmentioning

confidence: 99%

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Almost-Free E-Rings of Cardinality ℵ₁

Göbel¹,

Shelah²,

Strüngmann³

2003

Can. j. math.

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Abstract. An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R + is multiplication by some ring element. The existence of almost-free E-rings of cardinality greater than 2 ℵ 0 is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ℵ 1 ≤ λ ≤ 2 ℵ 0 we construct E-rings of cardinality λ in ZFC which have ℵ 1 -free additive structure. For λ = ℵ 1 we therefore obtain the existence of almost-free E-rings of cardinality ℵ 1 in ZFC.

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“…A first example of an ℵ 1 -free R-module M of size ℵ 1 with trivial dual was given in Eda [3]. Some years later we improved this result showing the existence of such modules with endomorphism ring R, see [2,7] or [8]. If we want to replace ℵ 1 by ℵ 2 or any higher cardinal, then we necessarily encounter additional set theoretic restriction, see [6].…”

Section: Introductionmentioning

confidence: 99%

$$\aleph_n$$ -Free Modules with Trivial Duals

Göbel

Shelah

2009

Results. Math.

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In the first part of this paper we introduce a simplified version of a new Black Box from Shelah [11] which can be used to construct complicated ℵn-free abelian groups for any natural number n ∈ N. In the second part we apply this prediction principle to derive for many commutative rings R the existence of ℵn-free R-modules M with trivial dual M * = 0, where M * = Hom(M, R). The minimal size of the ℵn-free abelian groups constructed below is n, and this lower bound is also necessary as can be seen immediately if we apply GCH. Mathematics Subject Classification (2000). Primary 13C05, 13C10, 13C13, 20K20, 20K25, 20K30; Secondary 03E05, 03E35.

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Indecomposable Almost Free Modules—The Local Case

Cited by 27 publications

References 25 publications

Closure properties of $\varinjlim\mathcal C$

Closure properties of $\varinjlim\mathcal C$

Almost-Free E-Rings of Cardinality ℵ₁

$$\aleph_n$$ -Free Modules with Trivial Duals

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Indecomposable Almost Free Modules—The Local Case

Cited by 27 publications

References 25 publications

Closure properties of $\varinjlim\mathcal C$

Closure properties of $\varinjlim\mathcal C$

Almost-Free E-Rings of Cardinality ℵ1

$$\aleph_n$$ -Free Modules with Trivial Duals

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Almost-Free E-Rings of Cardinality ℵ₁