On the Existence of Rigid ℵ1-Free Abelian Groups of Cardinality ℵ1

Göbel, Rüdiger; Shelah, Saharon

doi:10.1007/978-94-011-0443-2_18

Cited by 4 publications

(5 citation statements)

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“…This stimulated the question of posing additional algebraic conditions on M. In [12,13] we elaborate the 'case ℵ 1 ': If R is a countable ring with free additive structure, then there exists an ℵ 1 -free abelian group G of cardinality ℵ 1 with End G = R. There are also related results in [4], but restricted to ℵ 1 . Moreover, a natural barrier appears: the existence of indecomposable ℵ 2 -free groups of cardinality ℵ 2 or the existence of such groups with endomorphism ring Z is undecidable.…”

Section: Introductionmentioning

confidence: 99%

Prescribing endomorphism algebras of $\aleph_n$-free modules

Göbel¹,

Herden²,

Shelah³

2014

J. Eur. Math. Soc.

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors consider almost-free abelian groups and assume the existence of large canonical, free subgroups. Nevertheless, there exist ℵ 1 -separable torsion-free groups G of size ℵ 1 with a basic subgroup B of rank ℵ 1 such that all subgroups of G disjoint from B are also free, but the groups G are still not free. What else can we say about G? The other paper deals with Kaplansky's test problems (which are excellent indicators that the objects defy classification). The authors are able to construct very free abelian groups and verify the test problems for them by a careful choice of particular elements of their endomorphism rings.Accordingly, we want to investigate and construct ℵ n -free R-modules M (with n an arbitrary, but fixed natural number) over a domain R with End R M = R for the first time more systematically and uniformly. Recall that M is ℵ n -free if every subset of size < ℵ n is contained in a pure, free submodule of M. The requirement End R M = R implies that M is indecomposable, hence complicated. (We will also allow that End R M is a prescribed R-algebra, as in the title of this paper.)By now it is folklore to construct such modules M using additional set-theoretic axioms, most notably Jensen's ♦-principle. In this case the freeness condition can even be strengthened (see [6] and many examples in [9]). However, if we insist on proving this result in ordinary ZFC, then the known arguments fail: The classical constructions from the fundamental paper by Corner [2] do not apply because they are based on pure submodules of p-adic completions of free A-modules, which are never even ℵ 1 -free. If we use Shelah's Black Box instead of Jensen's ♦-principle, then the constructed modules M are still ℵ 1 -free, but always fail to be even ℵ 2 -free (see [4]). Thus we must develop new methods, which are presented for the first time in Sections 2 to 6, to achieve the desired result (Main Theorem 7.6). With these methods we provide a useful tool for a wide range of problems concerning ℵ n -free structures which can then be attacked.

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Section: Introductionmentioning

confidence: 99%

Prescribing endomorphism algebras of $\aleph_n$-free modules

Göbel¹,

Herden²,

Shelah³

2014

J. Eur. Math. Soc.

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“…We will construct G as an A-module and A is identified with endomorphisms acting by scalar multiplication. If A = R, we derive the existence of ℵ 1 -free R-modules of cardinality ℵ 1 with End G = R, a result about indecomposable R-modules known only in the case R = Z from our recent paper [21]. The main difficulty in passing from Z to R can be seen in the local case when R is a local ring with just one prime p, e.g.…”

Section: Main Corollary 31mentioning

confidence: 99%

“…This of course was due to other difficulties that had to be settled first. Assuming Martin's axiom (MA) together with ZFC and ℵ 2 < 2 ℵ 0 any ℵ 2 -free group G of cardinality < 2 ℵ 0 is separable and hence has endomorphism ring Z only in the trivial case when G = Z, see [21].…”

Section: Main Corollary 31mentioning

confidence: 99%

“…Infinitely many primes -by arithmetic -provide a rigid system (= modules with no homomorphisms = 0 between them). Hence homomorphisms can be restricted in their activity on G by building into G a rigid system in a suitable way [21]. Finally they 'calm down' to scalar multiplication on G. This is no longer possible in the local case.…”

Section: Introductionmentioning

confidence: 99%

“…We will show the next corollary which will follow immediately from our Main Theorem in Section 3.Main Corollary 3.1 Let A = 0 be a R-free R-algebra over a countable, principal ideal domain R which is not a field and let |A| < λ ≤ 2 ℵ 0 , then there exists an ℵ 1 -free R-module G of cardinality λ with End G = A.We will construct G as an A-module and A is identified with endomorphisms acting by scalar multiplication. If A = R, we derive the existence of ℵ 1 -free R-modules of cardinality ℵ 1 with End G = R, a result about indecomposable R-modules known only in the case R = Z from our recent paper [21]. The main difficulty in passing from…”

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Indecomposable Almost Free Modules—The Local Case

Göbel

Shelah

1998

Can. j. math.

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ABSTRACT. Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an @ 1 -free R-module G of rank @ 1 with endomorphism algebra End R G = A. Clearly the result does not hold for fields. Recall that an R-module is @ 1 -free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra A in use. For instance, if we choose A = R, then clearly G is an indecomposable 'almost free' module. The existence of such modules was unknown for rings with only finitely many primes like R = Z (p) , the integers localized at some prime p. The result complements a classical realization theorem of Corner's showing that any such algebra is an endomorphism algebra of some torsionfree, reduced R-module G of countable rank. Its proof is based on new combinatorialalgebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.

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