Rings on Abelian torsion-free groups of finite rank
نویسندگان
چکیده
منابع مشابه
On the complexity of the classification problem for torsion-free abelian groups of finite rank
In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-fre...
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Let RA be a group ring of an abelian group A and let I be an ideal of RA . We say that a subgroup B of A controls I if I = (I ∩ RB)RA. The intersection c(I) of all subgroups of A controlling I is said to be the controller of the ideal I ; c(I) is the minimal subgroup of A which controls the ideal I . The ideal I is said to be faithful if I = A ∩ (1 + I) = 1. In theorem 4 we consider some method...
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Let G be a torsion-free abelian group of type (0, 0, 0, . . . ) and R an integrally closed integral domain with quotient field K. We show that every divisorial ideal (respectively, t-ideal) J of the group ring R[X;G] is of the form J = hIR[X;G] for some h ∈ K[X;G] and a divisorial ideal (respectively, t-ideal) I of R. Consequently, there are natural monoid isomorphisms Cl(R) ∼= Cl(R[X;G]) and C...
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Suppose that P is a convex polyhedron in the hyperbolic 3-space with finite volume and P has integer ( > 1) submultiples of it as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group G associated to P is one, then P has exactly one ideal vertex of type (2,2,2,2) and G has an index two subgroup which does not cont...
متن کاملBorel superrigidity and the classification problem for the torsion-free abelian groups of finite rank
In 1937, Baer solved the classification problem for the torsion-free abelian groups of rank 1. Since then, despite the efforts of many mathematicians, no satisfactory solution has been found of the classification problem for the torsion-free abelian groups of rank n ≥ 2. So it is natural to ask whether the classification problem for the higher rank groups is genuinely difficult. In this article...
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ژورنال
عنوان ژورنال: Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
سال: 2021
ISSN: 0138-4821,2191-0383
DOI: 10.1007/s13366-021-00585-0