Torsion-free Abelian Groups Research Articles

Definability of Abelian groups by homomorphism groups and endomorphism semigroups

In the paper, the isomorphism problem for completely decomposable Abelian torsion-free groups of finite rank is treated under the assumption that the groups of homomorphisms of these groups into some Abelian group are isomorphic and, moreover, the endomorphism semigroups of the groups are isomorphic.

RUELLE TYPE L-FUNCTIONS VERSUS DETERMINANTS OF LAPLACIANS FOR TORSION FREE ABELIAN GROUPS

We study Ruelle's type zeta and L-functions for a torsion free abelian group Γ of rank ν ≥ 2 defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when ν = 2, 4 and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus ℝν/Γ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

The isomorphism problem for torsion-free Abelian groups is analytic complete

We prove that the isomorphism problem for torsion-free Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete.

Determination of a class of countable-rank, torsion-free Abelian groups by their endomorphism rings

Determination, up to near isomorphism, of countable-rank, block-rigid, local, almost completely decomposable groups of ring type with cyclic regulator quotient by their endomorphism rings in this class has been proved.

The test rank of a soluble product of free Abelian groups

We consider the variety of all soluble groups of derived length at most , . Suppose that a finitely generated group is a free product in the variety of Abelian torsion-free groups. It is proved that the test rank of is one less than the number of factors. A test set of elements is written out explicitly.Bibliography: 27 titles.

Finite index supergroups and subgroups of torsionfree abelian groups of rank two

Every torsionfree abelian group A of rank two is a subgroup of Q ⊕ Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2 × 2 -matrices as the bonding maps. Using these direct systems we classify all subgroups of Q ⊕ Q which are finite index supergroups of A or finite index subgroups of A. Using this classification we prove that for each prime p there exists a torsionfree abelian group A satisfying the following, where A ⩽ Q ⊕ Q and all supergroups are subgroups of Q ⊕ Q : (1) for each natural number s there are ∑ q | s , gcd ( p , q ) = 1 q s-index supergroups and also ∑ q | s , gcd ( p , q ) = 1 q s-index subgroups; (2) each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index subgroups are non-isomorphic.

A THEOREM ON MATRIX GROUPS

Let K be a principal ideal domain, and An, with n ≥ 3, be a finitely generated torsion-free abelian group of rank n. Let Ω be a finite subset of KAn\{0} and U(KAn) the group of units of KAn. For a multiplicative monoid P generated by U(KAn) and Ω, we prove that any generating set for [Formula: see text] contains infinitely many elements not in [Formula: see text]. Furthermore, we present a way of constructing elements of [Formula: see text] not in [Formula: see text] for n ≥ 3. In the case where K is not a field the aforementioned results hold for n ≥ 2.

A non-perfect surjective cellular cover of PSL(3,F(t))

Let F(t) be the function field in one variable over the finite field F. We construct a surjective cellular cover γ : 𝒢 → PSL(3, F(t)), where 𝒢 = G ○ E, G = St(3, F(t)), E = Ext(ℚ/ℤ K͂2(F(t))) and G ○ E is the commuting product with G ∩ E = K͂2(F(t)). Here K͂2(F(t)) is the kernel of St(3, F(t)) ↠ PSL(3, F(t)). Since 𝒢/[𝒢, 𝒢] ≅ E/K͂2(F(t)) is a nontrivial torsion free divisible abelian group, this gives a negative answer to a question raised in the paper “Cellular covers of groups” (J. Pure and Applied Algebra 208 (2007)), by E. Farjoun, R. Göbel and the author. We asked whether a surjective cellular cover of a perfect group is perfect.

The entropy of algebraic actions of countable torsion-free abelian groups

This paper is concerned with the entropy of an action of a countable torsion-free abelian group $G$ by continuous automorphisms of a compact abelian group $X$. A formula is obtained that expresses the entropy in terms of the Mahler measure of a great

Invariant decomposition of functions with respect to commuting invertible transformations

Consider a 1 ,..., , an ∈ R arbitrary elements. We characterize those functions f: R→ R that decompose into the sum of a j -periodic functions, i.e., f=f 1 +---+f n with Δ aJ f(x):= f(x+aj)-f(x) = 0. We showthat f has such a decomposition if and only if for all partitions Β 1 UB 2 U...US N = {a 1 ,..., a n ) } with Bj consisting of commensurable elements with least common multiples b j one has Δ b1 ... Δb N f = 0. Actually, we prove a more general result for periodic decompositions of functions f: A→ R defined on an Abelian group A; in fact, we even consider invariant decompositions of functions f: A → R with respect to commuting, invertible self-mappings of some abstract set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real-valued periodic decomposition of an integer-valued function implies the existence of an integer-valued periodic decomposition with the same periods.

A Duality for Self-Slender Modules

The constructibility axiom, V = L, is a statement in mathematics that is independent of the ZFC axioms of set theory. One of its consequences is (μ) = measurable cardinals do not exist. We assume (μ). Let R be a ring, and let G be a right R-module. We say that G is slender if for each indexed set {R i | i ∈ ℐ} (of nonmeasurable cardinality) such that R ≅ R i for each i ∈ ℐ, the canonical map is an isomorphism. Reduced countable torsion-free abelian groups are slender groups. Nonzero slender injective right R-modules do not exist. We say that G is self-slender if for each (nonmeasurable) cardinal ℵ, the canonical map is an isomorphism. Reduced countable torsion-free abelian groups G are self-slender. We show in this article that if we assume (μ), then for each self-slender right R-module G there is a category G-coPlex in which G is a slender injective object and such that End(G)-Mod is dual to G-coPlex. Thus there are many nonzero slender injective objects.

Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ⊗ E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in 𝒢, the class of self-small mixed groups G with G/tG≅ ℚ n for some n < ω. We obtain a new characterization of the elements of 𝒢, and demonstrate that 𝒢 differs in various ways from the class 𝒯 ℱ of torsion-free abelian groups of finite rank despite the fact that the quasi-category ℚ 𝒢 is dual to a full subcategory of ℚ 𝒯 ℱ.

Representations of Finite Posets Over Discrete Valuation Rings

Given a discrete valuation ring R, a non-negative integer m, and a finite partially ordered set S, the representation type of a category rep(S*, R, m) of finite rank free R-module representations of S is characterized in terms of S and m. As an application, representation types of some categories of torsion-free abelian groups of finite rank are determined.

Settings for a study of finite rank Butler groups

A finite rank Butler group G is a torsionfree Abelian groups that is the sum of m rank one subgroups; G is a B ( n ) -group if n is the maximum number of independent relations between the m subgroups. After the well-known class B ( 0 ) , the much studied B ( 1 ) and the first approaches to B ( 2 ) , in this paper we generalize some of the tools used before and introduce new ones to work in every B ( n ) . We study some of the relationships between these tools, and while clarifying some basic settings describe an interesting class of indecomposables.

Quasi-localizations of ℤ

Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has \(R \subseteq M \subseteq \mathbb{Q} \otimes _\mathbb{Z} R\). We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R+, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family \(\mathcal{F}\) of submodules and thus \((M,\mathcal{F})\). We will restrict our discussion to the case M = R such that \((R,\mathcal{F}) = R\), and in this case we call the family \(\mathcal{F}\) of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank.

The Chase radical and reduced products

Let κ be a regular uncountable cardinal. We shall give a criterion for certain reduced products ∏α<κGα/∏α<κ<Gα of torsion-free abelian groups Gα(α<κ) to be ℵ1-free. As an application we shall show that the norm of the Chase radical is ℵ1 in ZFC, a result which was previously known only under the assumption of the continuum hypothesis 2ℵ0=ℵ1.

Near Isomorphism for a Class of Infinite Rank Torsion-Free Abelian Groups

The notion of “near isomorphism” for torsion-free Abelian groups of finite rank is well known. In particular, this concept turned out to be of importance for classifying almost completely decomposable groups. We extend near isomorphism to classes of torsion-free Abelian groups of infinite rank which are unions of bcd–groups, this is to say unions of groups which are bounded essential extensions of completely decomposable groups. Moreover, we show that nearly isomorphic groups of this class also have nearly isomorphic endomorphism rings considered as Abelian groups.

Turing degrees of isomorphism types of algebraic objects

The Turing degree spectrum of a countable structure 𝒜 is the set of all Turing degrees of isomorphic copies of 𝒜. The Turing degree of the isomorphism type of 𝒜, is the least Turing degree in its degree spectrum. We show that there are structures with isomorphism types of arbitrary Turing degrees in each of the following classes: countable fields, rings, and torsion-free Abelian groups of any finite rank. We also show that there are structures in each of these classes the isomorphism types of which do not have Turing degrees. The case of torsion-free Abelian groups of finite rank settles a question left open by Knight, Downey and Jockusch [Downey, Complexity, logic, and recursion theory, Lecture Notes in Pure and Applied Mathematics 187 (ed. A. Sorbi; Marcel Dekker, New York, 1997) 157–205].

Constructing sharply transitive R-modules of rank ⩽

In this note we will give an elementary proof of the existence of sharply transitive R-modules M over principal ideal domains R. An R-module is sharply transitive (or a UT-module) if its R-automorphism group acts sharply transitively on the pure elements of M. We will assume that M is torsion-free; thus pure elements are simply those elements divisible only by units of R in M. We provide examples of UT-modules of rank ⩽ , while the existence of UT-modules of rank ⩾ was shown recently in Göbel and Shelah [R. Göbel and S. Shelah. Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Applied Math. 236 (Marcel Dekker, 2004), pp. 271–290.] using the more complicated machinery of prediction principles. The existence of countable abelian UT-groups, which follows from this note, was left open in earlier works. Here we require and exploit the existence of algebraically independent elements over the base ring R. (Thus we will need |R| < .) First we will convert the UT problem on modules (as suggested in Herden [D. Herden. Uniquely transitive R-modules. Ph.D. thesis. University of Duisburg-Essen, Campus Essen (2005).]) into a problem on suitable R-algebras. This reduces its solution to a few simple steps and makes the proofs more transparent, requiring only basic results in module theory.

Formula omitted]-algebras that are sharply transitive modules

We will consider modules over principal ideal domains that are neither fields nor complete discrete valuation rings and show their wild behavior in two instances, which we explain for simplicity in the category of abelian groups. In the first case we consider constructions of large E-rings: This question was answered in 1987 by Dugas, Mader, Vinsonhaler [M. Dugas, A. Mader, C. Vinsonhaler, Large E-rings exist, J. Algebra 108 (1) (1987) 88–101]. Recall that a ring A is an E-ring if the canonical map End Z A → A ( φ ↦ φ ( 1 ) ) is an isomorphism. The second question was raised in 1997 by Emmanuel Dror Farjoun and answered recently in [R. Göbel, S. Shelah, Uniquely transitive torsion-free abelian groups, in: Rings, Modules, Algebras, and Abelian Groups, in: Pure Appl. Math., vol. 236, Marcel Dekker, 2004, pp. 271–290]. There is a (non-trivial) torsion-free abelian group G (other than Z and with pure elements) such that its automorphism group Aut G acts sharply transitive on the pure elements of G. These abelian groups G are called UT-groups (or UT-modules for more general rings). Recall that pure elements of a torsion-free abelian group are those divisible only by ±1 and note that automorphisms leave the family p G of pure elements of G invariant. The two apparently distinct questions, can be unified and strengthened. The link between them is the notion of a pure-invertible ring A, which simply means that all pure elements of A Z (seen as abelian group) are units of the commutative ring A. For technical reasons (to have non-trivial pure elements) we also require that A is ℵ 1 -free as abelian group. In the first part of this paper we transform Farjoun's problem into a ring theoretic question on A. It turns out that A must be a pure-invertible ring and an E-ring at the same time. In the second part of the paper we construct such rings assuming ZFC + CH (the special continuum hypothesis). Thus the existence of UT-groups follows and as a byproduct we obtain a new class of E-rings which are at the same time principal ideal domains.