Finite Rank Torsion Free Abelian Groups and Rings

Arnold, David M.

doi:10.1007/bfb0094245

Cited by 269 publications

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“…As a consequence of Theorem 6, we obtain similar results for normal subgroup lattices. The proofs for c) and d) are similar, using this time the fact that Z and Q are cancellable from direct products of Abelian groups (see [1,Corollary 8.8]). p REMARK 10.…”

Section: E W Jv(b) : V(b) 3 V(w(b)) Is An Isomorphism) Then V(w(b))mentioning

confidence: 97%

Commutativity Criterions Using Normal Subgroup Lattices

Breaz¹

2009

Rend. Sem. Mat. Univ. Padova

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-We prove that a group G is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of G is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group B such that the normal subgroup lattice of B Â G is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group A can be determined in the class of all groups by the lattice of all normal subgroups of some groups, e.g. if A is an Abelian group and G is a group such that Z Â A and Z Â G have isomorphic normal subgroup lattices then A and G are isomorphic groups.

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Section: E W Jv(b) : V(b) 3 V(w(b)) Is An Isomorphism) Then V(w(b))mentioning

confidence: 97%

Commutativity Criterions Using Normal Subgroup Lattices

Breaz¹

2009

Rend. Sem. Mat. Univ. Padova

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“…. , (X h(X ) ), and by arguing above with Warfield's theorem [1,Theorem 13.9], there are integers m, m 1 , . .…”

Section: Proof (1) If and Only If (2) Let G ∈ (E) Then H(g)mentioning

confidence: 99%

Diagrams of an Abelian Group

Faticoni¹

2009

Bull. Aust. Math. Soc.

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In this paper, we characterize quadratic number fields possessing unique factorization in terms of the power cancellation property of torsion-free rank-two abelian groups, in terms of -unique decomposition, in terms of a pair of point set topological properties of Eilenberg-Mac Lane spaces, and in terms of the sequence of rational primes. We give a complete set of topological invariants of abelian groups, we characterize those abelian groups that have the power cancellation property in the category of abelian groups, and we characterize those abelian groups that have -unique decomposition. Our methods can be used to characterize any direct sum decomposition property of an abelian group.2000 Mathematics subject classification: primary 20K99; secondary 16K99.

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“…For example, it is well known that if T is a right .R-module and if To ®H A = 0 for each finitely generated .R-submodule T o C T then T® R A -0. In 3.16 and 3.17 we construct a countable local commutative integral domain R and a countable .R-module A such that (1) R -End (A), (2) To ®RA ^ 0 for each nonzero finitely generated (respectively finitely presented) .R-module To, and (3) T ®R A = 0 for some nonzero (respectively nonzero finitely generated) .R-module T. [3] Torsion-free Abelian groups 179…”

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

confidence: 99%

“…Our notation and terminology follow [3] and [15], and information on the finite topology can be found in [15]. As usual, Z is the ring of rational integers, Q is the field of rational numbers, and given a torsion-free group G, QG is the divisible hull of G. We identify QG = Q ® Z G .…”

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

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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Finite Rank Torsion Free Abelian Groups and Rings

Cited by 269 publications

References 0 publications

Commutativity Criterions Using Normal Subgroup Lattices

Commutativity Criterions Using Normal Subgroup Lattices

Diagrams of an Abelian Group

Torsion-free Abelian groups torsion over their endomorphism rings

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