Almost Free Modules Revisited

Eklof, Paul C.; Mekler, Alan H.

doi:10.1016/s0924-6509(02)80009-5

Cited by 59 publications

(51 citation statements)

References 202 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In L, such a group is known to exists at precisely the non-weakly compact, regular cardinals; see Eklof and Mekler's book [EM02]. Combining these two facts, we have our proof.…”

Section: Further Workmentioning

confidence: 62%

Tameness From Large Cardinal Axioms

Boney¹

2014

J. symb. log.

View full text Add to dashboard Cite

Abstract. We show that Shelah's Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ-tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly from large cardinals.

show abstract

“…In L, such a group is known to exists at precisely the non-weakly compact, regular cardinals; see Eklof and Mekler's book [EM02]. Combining these two facts, we have our proof.…”

Section: Further Workmentioning

confidence: 62%

Tameness From Large Cardinal Axioms

Boney¹

2014

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…The passage through R ω -modules has the great advantage that the proofs become very transparent essentially using a few 'linear algebra' arguments also accessible for graduate students. The result closes a gap of Eklof and Shelah (1999) and Eklof and Mekler (2002), provides a good starting point for Fuchs and Göbel, and gives a new construction of indecomposable modules in general using a counting argument. …”

mentioning

confidence: 57%

“…The corollary on the existence of large absolutely (fully) rigid abelian groups replaces the earlier unsuccessful approach in [12] and [11,Chapter XV]: Let R = 0 be any fixed countable ring. Then by Corollary 4.2 there exists an absolutely rigid R ω -module of size λ (or an absolute family of size λ of non-trivial R-modules with only the zero-homomorphism between distinct members) iff λ < κ(ω).…”

Section: Introductionmentioning

confidence: 96%

“…Thus as a byproduct we present a new construction of large, absolutely indecomposable abelian groups, not using stationary sets as in [25,7]. So, if we restrict ourselves to the problem on the existence of large absolute indecomposable abelian groups addressed in [12,11], then it follows from the above (realizing for example Z as the endomorphism ring in Corollary 4.2) that from λ < κ(ω) follows the existence of such abelian groups. The converse direction would need a strengthening of Theorem 2.2 from [12] now showing the existence of non-trivial idempotents.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Absolutely indecomposable modules

Göbel

Shelah

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about R-modules over a large class of commutative rings R with endomorphism ring R which remains the same when passing to a generic extension of the universe. It turns out that 'large' in this context has a precise meaning, namely being smaller than the first ω-Erdős cardinal defined below. We will first apply a result on large rigid valuated trees with a similar property established by Shelah in 1982, and will prove the existence of related 'R ω -modules' (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through R ω -modules has the great advantage that the proofs become very transparent essentially using a few 'linear algebra' arguments also accessible for graduate students. The result closes a gap of Eklof and Shelah (1999) and Eklof and Mekler (2002), provides a good starting point for Fuchs and Göbel, and gives a new construction of indecomposable modules in general using a counting argument.

show abstract

“…We now translate a notion that has been of considerable importance in the study of (torsion-free) free groups (see [10]) into the language of valuated vector spaces. We will say the valuated vector space V is ℵ 1 -coseparable if it is separable (i.e., V (ω) = {0}), and for all subspaces W of V with V /W countable there is a closed subspace U of V such that U ⊆ W and V /U is countable -note that this implies that the quotient valuated vector space V /U is separable and hence free, so that V is isometric to U ⊕ F where F is countable and free.…”

Section: Totally Projective Groups and Generalizationsmentioning

confidence: 99%

Nice elongations of primary abelian groups

Danchev¹,

Keef²

2010

Publicacions Matemàtiques

View full text Add to dashboard Cite

Suppose N is a nice subgroup of the primary abelian group G and A = G/N . The paper discusses various contexts in which G satisfying some property implies that A also satisfies the property, or visa versa, especially when N is countable. For example, if n is a positive integer, G has length not exceeding ω 1 and N is countable, then G is n-summable iff A is n-summable. When A is separable and N is countable, we discuss the condition that any such G decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that A must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups. Introduction and TerminologyBy the term "group" we will mean an abelian p-group, where p is a prime fixed for the duration. Our group theoretic terminology and notation will generally follow [11]. We will say a group G is Σ-cyclic if it is isomorphic to a direct sum of cyclic groups. We will also utilize the language of valuated groups and valuated vector spaces (see [22] and [12], respectively); for example, if Y is a valuated group, the letter v will be reserved for its valuation, and for any ordinal α, by Y (α) we will mean the subgroup {y ∈ Y : v(y) ≥ α}. We will implicitly assume that all valuated vector spaces are over Z p .If 0 → X → G → A → 0 is a short exact sequence, we will routinely identify X with an actual subgroup of G and A with the quotient group G/X; we will say that G is an elongation of A by X. We then say that G is a nice-elongation of A if X is a nice subgroup (i.e., for every y ∈ G, the coset y + X has an element of maximum height). Further, 2000 Mathematics Subject Classification. 20K10.

show abstract

Almost Free Modules Revisited

Cited by 59 publications

References 202 publications

Tameness From Large Cardinal Axioms

Tameness From Large Cardinal Axioms

Absolutely indecomposable modules

Nice elongations of primary abelian groups

Contact Info

Product

Resources

About