Torsion-free Abelian Groups Of Rank Research Articles

The classification of non-commutative torsion-free rings of rank two

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified.

Rank-two solenoidal endomorphisms

Let $G$ be a torsion-free abelian group of rank two and let $\phi$ be an endomorphism of $G$, called a rank-two \emph{solenoidal endomorphism}. Then it is represented by a $2\times 2$-matrix $M_\phi$ with rational entries. The purpose of this article is to prove the following: The group, $\mathrm{coker}(\phi)$, of the cokernut of $\phi$ is finite if and only if $M_\phi$ is nonsingular, and if it is so, then we give an explicit formula for the order of $\mathrm{coker}(\phi)$, $[G:\mathrm{im}(\phi)]$, in terms of $p$-adic absolute values of the determinant of $M_\phi$. Since $G$ is abelian, the Reidemeister number of $\phi$ is equal to the order of the cokernut of $\mathrm{id}-\phi$ and, when it is finite, it is equal to the number of fixed points of the Pontryagin dual $\widehat\phi$ of $\phi$. Thereby, we solve completely the problem raised in \cite{Miles} of finding the possible sequences of periodic point counts for \emph{all} endomorphisms of the rank-two solenoids.

New examples of indecomposable torsion-free abelian groups of finite rank and rings on them

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative.

On the Residual Finiteness of a Class of Infinite Soluble Groups

Let [Formula: see text] be a completely decomposable homogeneous torsion-free abelian group of rank [Formula: see text] ([Formula: see text]). Let [Formula: see text] be the split extension of [Formula: see text] by an automorphism [Formula: see text] which is a cyclic permutation of the direct components twisted by a rational integer [Formula: see text]. Then [Formula: see text] is an infinite soluble group. In this paper, the residual finiteness of [Formula: see text] is investigated.

A note on Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups in the context of additive groups of rings of finite rank

In the class of torsion-free abelian groups of finite rank, Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups is reduced to indecomposable groups. Torsion-free abelian groups $A$ of rank two such that every associative ring on $A$ is commutative but there exists a~non-commutative ring with the additive group $A$, are classified. Moreover, several valuable results concerning rings on torsion-free abelian groups of rank two achieved by Beaumont, Wisner, Jackett, Aghdam and Najafizadeh are complemented and their proof are greatly simplified.$

Structure of Finite-Dimensional Protori

A Structure Theorem for Protori is derived for the category of finite-dimensional protori (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces.

On the Nil R-mod Abelian Groups

Let M be an abelian group and N be a subgroup of M. Given a ring R, the notion of a left nil R-mod group modulo N is introduced as a generalization of a left nil R-mod group. Moreover, we define \(\text {Nil}^{l}_{R}(M)\) as a subgroup of M which is defined to be the intersection of all subgroups N of M such that M is a left nil R-mod group modulo N. In this paper, we investigate some properties of the \(\text {Nil}^{l}_{R}(M)\). Moreover, we describe it in some torsion-free abelian groups of rank two.

A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

AbstractA hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group A, the quotient group modulo the square subgroup of A is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group A, the square subgroup of A considered in the class of associative rings, is a characteristic subgroup of A.

On the Quasi-Endomorphism Rings of Quasi-Decomposable Torsion-Free Abelian Groups of Rank 4 with a Strongly Indecomposable Quasi-Summand of Rank 2

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups G of rank 4 that are quasi-decomposable into a direct sum of groups A1 and A2 of rank 1 and a strongly indecomposable group B of rank 2 in the case where the quasi-homomorphism group ℚ ⨂ Hom(B,A2) has rank 2.

Classifications of Computable Structures

Let K be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from K such that every structure in K is isomorphic to exactly one structure on the list. Such a list is called a computable classification of K, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a 0'-oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank 1, even though these families are both closely allied with computable algebraic fields.

Direct decomposition theory under near-isomorphism for a class of infinite rank torsion-free abelian groups

Abstract Near-isomorphism is known as the right concept for classification theorems in the theory of almost completely decomposable groups. As a natural generalization the authors extended in [6] the notion of near-isomorphism to Abelian groups of arbitrary rank. In this article we investigate non-isomorphic direct decompositions of a class of infinite rank torsion-free abelian groups which were defined in [4] as special epimorphic images of so-called almost rigid groups. A complete classification of such decompositions up to near-isomorphism is given.

一类有限秩Abel群的自同构

本文给出了一类有限秩的具有C2自同构的无挠Abel群。 In this paper, we mainly study a class of finite rank torsion free abelian groups which their automorphism groups are C2.

On the square subgroups of decomposable torsion-free abelian groups of rank three

AbstractThe square subgroup of an abelian group

Quasi-endomorphism rings of almost completely decomposable torsion-free Abelian groups of rank 4 that do not coincide with their pseudo-socles

We obtain a description of quasi-endomorphism rings of almost completely decomposable torsion-free Abelian groups of rank 4 that do not coincide with their pseudo-socles.

Quasi-Endomorphism Rings of Almost Completely Decomposable Torsion-Free Abelian Groups of Rank 4 With Zero Jacobson Radical

In this paper, a description of rings of quasi-endomorphisms of almost completely decomposable torsion-free Abelian groups of rank 4 with zero Jacobson radical is obtained.

Quasi-Endomorphism Rings of Strongly Indecomposable Torsion-Free Abelian Groups of Rank 4 With Pseudosocles of Rank 1

We obtain the matrix representation of the minimal rational algebra whose subalgebras, up to isomorphism, are quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 4 with pseudosocles of rank 1.

On the indecomposable torsion-free abelian groups of rank two

On the indecomposable torsion-free abelian groups of rank two

On the ideals of torsion-free rings of rank one and two

Let A be a torsion-free Abelian group of rank one or two. We use the type set of A to give necessary and sufficient conditions for the subgroups of A to be ideals in every ring on A.

Rings of quasi-endomorphisms of strongly indecomposable torsion-free abelian groups of rank 4 with pseudosocles of rank 3

In this paper, the classification of rings of quasi-endomorphisms of strongly indecomposable Abelian torsion-free groups of rank 4 with pseudosocles of rank 3 is obtained.

The classification problem for S-local torsion-free abelian groups of finite rank

Suppose that n ⩾ 2 and that S, T are sets of primes. Then the classification problem for the S-local torsion-free abelian groups of rank n is Borel reducible to the classification problem for the T-local torsion-free abelian groups of rank n if and only if S ⊆ T .