On Endomorphism Rings of Primary Abelian Groups

Corner, A. L. S.

doi:10.1093/qmath/20.1.277

Cited by 88 publications

(35 citation statements)

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“…The present constructions will clone the elegant technique used in [7] to construct the short exact sequence (1.1).…”

Section: Is a Direct Sum Of Cyclic R-submodules Of M^°\ The Direct mentioning

confidence: 99%

“…Assume without loss of generality that 7n©n E E. As in [7], substitute (2) and (3) The rest of the proof is different enough from [7, p.67-68] that we give a more detailed account. However, the idea is the same: realise a given endomorphism 77: A -> A as left multiplication by the limit of a Cauchy net (sequence) in R.…”

Section: Relativised Finite Topologies Given a Ring S And A Leftmentioning

confidence: 99%

“…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%

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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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“…The present constructions will clone the elegant technique used in [7] to construct the short exact sequence (1.1).…”

Section: Is a Direct Sum Of Cyclic R-submodules Of M^°\ The Direct mentioning

confidence: 99%

Section: Relativised Finite Topologies Given a Ring S And A Leftmentioning

confidence: 99%

“…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%

Section: Introductionmentioning

confidence: 99%

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Torsion-free Abelian groups torsion over their endomorphism rings

Faticoni¹

1994

Bull. Austral. Math. Soc.

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“…2* An extension of Corner's Theorem* Given a torsion complete p-group C with unbounded basic subgroup B of cardinality not more than 2*°, Corner's theorem [2,Th. 2.1] gives sufficient conditions on a subring R of E{B) for there to exist a pure subgroup G of C containing B such that E(G) = E S {G)@R. It is necessary for our example in section 4 to weaken slightly the conditions on the ring R. In doing this we obtain a group G such that E(G) = E S (G) + R without neces-…”

Section: On the Endomorphism Ring Of An Abelian ί-Group And Of A Larmentioning

confidence: 99%