Prescribing Endomorphism Algebras, a Unified Treatment

Corner, A. L. S.; Göbel, Rüdiger

doi:10.1112/plms/s3-50.3.447

Cited by 153 publications

(172 citation statements)

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“…(A similar result, but replacing ω in (i) by arbitrary uncountable, regular cardinals follows from [3], see also [9].) Since P = n∈ω Ze n is ℵ 1 -free it is clear that the rank-condition in (ii) can be replaced by the requirement that Im ϕ is finitely generated (and free).…”

Section: Relating G(l) and Mclain-groupsmentioning

confidence: 85%

Stabilizers of direct composition series

Droste

Göbel

2009

Algebra Univers.

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Section: Relating G(l) and Mclain-groupsmentioning

confidence: 85%

Stabilizers of direct composition series

Droste

Göbel

2009

Algebra Univers.

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“…By 2.1 rrn -VETTI = t](m) for each finite set E C QM and each TO G E. Thus [J; -f](M) = 0. Inasmuch as A C M is a reduced group, we can proceed as usual (see for example [6,7,8,13,14,15]…”

Section: Proof: By 272 a Is An O(m)-submodule Of M Inasmuch As Ta mentioning

confidence: 99%

“…Inspection of the proof reveals that the group A is an extension of R by a free QR-module. In [7], the free module is replaced by a direct sum of cyclic modules that are discrete in a complete Hausdorff linear topology on R. Other realisation Theorems show that each cotorsion-free ring R is the endomorphism ring of a torsion-free group A, [8], and again A is an extension of a free .R-module by a free Q.R-module. These constructions do not allow us to vary some of the subtler End(.A)-module structure of A.…”

Section: Introductionmentioning

confidence: 99%

Torsion-free Abelian groups torsion over their endomorphism rings

Faticoni¹

1994

Bull. Austral. Math. Soc.

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We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End {A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z [X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal / in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable it-module A such that (1) R = End(A), (2) T o ®R A ^ 0 for each nonzero finitely generated (respectively finitely presented) fl-module To, but (3) T ®H A = 0 for some nonzero (respectively nonzero finitely generated) .R-module T.

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“…If R = Z, we derive the existence of ℵ 1 -free abelian groups of cardinality ℵ 1 , a result which was unknown. If Γ is any abelian semigroup, then we use Corner's ring R Γ , implicitly discussed in Corner, Göbel [4], and constructed for particular Γ s in [3] with special idempotents (expressed below), with free additive group and |R Γ | = max{|Γ|, ℵ 0 }. If |Γ| < 2 ℵ0 , we may apply the main theorem and find a family of ℵ 1 -free abelian groups G α (α ∈ Γ) of cardinality ℵ 1 such that for α, β ∈ Γ,…”

Section: Introductionmentioning

confidence: 99%