Torsion-free Groups Of Finite Rank Research Articles

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

Multiplications on torsion-free groups of finite rank

Multiplications on torsion-free groups of finite rank

Rings on Abelian torsion-free groups of finite rank

In the class of reduced Abelian torsion-free groups G of finite rank, we describe TI-groups, this means that every associative ring on G is filial. If every associative multiplication on G is the zero multiplication, then G is called a \(nil_a\)-group. It is proved that a reduced Abelian torsion-free group G of finite rank is a TI-group if and only if G is a homogeneous Murley group or G is a \(nil_a\)-group. We also study the interrelations between the class of homogeneous Murley groups and the class of \(nil_a\)-groups. For any type \(t\ne (\infty ,\infty ,\ldots )\) and every integer \(n>1\), there exist \(2^{\aleph _0}\) pairwise non-quasi-isomorphic homogeneous Murley groups of types t and \(n\in \mathbb {N}\) and rank n which are \(nil_a\)-groups. We describe types t and \(n\in \mathbb {N}\) such that there exists a homogeneous Murley group of type t and rank n which is not a \(nil_a\)-group.

Free subgroups with torsion quotients and profinite subgroups with torus quotients

Here “group” means abelian group. Compact connected groups contain \delta -subgroups, that is, compact totally disconnected subgroups with torus quotients, which are essential ingredients in the important Resolution Theorem, a description of compact groups. Dually, full free subgroups of discrete torsion-free groups of finite rank are studied in order to obtain a comprehensive picture of the abundance of \delta -subgroups of a protorus. Associated concepts are also considered.

Centrally essential torsion-free rings of finite rank

It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful.

Centrally essential endomorphism rings of abelian groups

We study Abelian groups A with centrally essential endomorphism ring If A is a such group which is either a torsion group or a non-reduced group, then the ring is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings.

Self-pure-generators over Dedekind domains

We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind domain are A-generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that type(B) is the inner type of A. This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.

A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ℤ [p−1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.

Local Abelian Torsion-Free Groups

A representation for p-local Abelian torsion-free groups of finite rank is obtained in terms of homomorphisms of p-adic modules of finite rank with fixed basis.

Direct decompositions of mixed abelian groups

We consider a sufficiently large subcategory of the category of mixed abelian groups of finite torsion-free rank and its quotient catego ry obtained by annihilating those homomorphisms which factor through the torsion. We prove that the second category provides a good approximation to the first category, but is much simpler: the groups of morphisms in the second category are finite rank torsion-free groups. This renders it possible to exam ine direct decompositions of mixed groups by the same methods as in the case of finite rank torsion-free groups.

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.

The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ℕ. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.

Class number of an abelian group

The groups in this paper are abelian. In this Addendum to [T.G. Faticoni, The class number of an abelian group, J. Algebra 314 (2007) 978–1008] we show that a problem in the direct sum decompositions of torsion-free finite rank groups implies several important problems of number theory.

Cancellation properties for quotient divisible groups

We investigate cancellation, the unit lifting property, and 1 in the stable range for mixed self-small abelian groups of finite torsion-free rank. Some of our results generalize results previously obtained for torsion-free finite rank groups by Fuchs, Kravchenko, Loonstra, and Stelzer. In particular, we prove Stelzer's main cancellation result for the class of mixed quotient divisible abelian groups.

The class number of an abelian group

The groups in this paper are abelian. Let G be a reduced torsion-free finite rank group. Then G is cocommutative if End ( G ) is commutative modulo the nil radical. The class number of G, h ( G ) , is the number of isomorphism classes of groups H that are locally isomorphic ( = nearly isomorphic) to G. We say that G satisfies the power cancellation property if G n ≅ H n for some group H and integer n > 0 implies that G ≅ H . We say that G has a Σ-unique decomposition if G n has a unique direct sum decomposition for each integer n > 0 . Let P o ( G ) = { groups H | H ⊕ H ′ = G m for some group H ′ and some integer m > 0 } . We say that G has internal cancellation if given H , K , L ∈ P o ( G ) such that H ⊕ K ≅ H ⊕ L then K ≅ L . We use the class number to study the torsion-free finite rank groups G that have the power cancellation property, or a Σ-unique decompositions, or the internal cancellation property. Furthermore, we show that the power cancellation property for cocommutative strongly indecomposable reduced torsion-free finite rank groups is equivalent to the problem of determining the class number of an algebraic number field. Let G ¯ be the integral closure of G. Using the Mayer–Vietoris sequence we show that there are finite groups associated with G of orders L ( p ) , m ˆ p , and n ˆ p such that h ( G ) = h ( G ¯ ) m ˆ p L ( p ) n ˆ p . Let E ( p ) = Z + p E ¯ where p is a rational prime and E ¯ is the ring of algebraic integers in the algebraic number field k = Q E ¯ . Let G ( p ) be a group such that End ( G ( p ) ) = E ( p ) . We show that the sequence { L ( p ) h ( G ( p ) ) / h ( G ¯ ( p ) ) | primes p } is asymptotically equal to the sequence { p f − 1 | primes p } where f = [ k : Q ] . Furthermore, for quadratic number fields k, h ( k ) = 1 iff { L ( p ) h ( G ( p ) ) | primes p } is asymptotically equal to the sequence of rational primes. This connects unique factorization in number fields with the sequence of rational primes, and with direct sum properties of integrally closed cocommutative strongly indecomposable torsion-free finite rank groups.

On the controllers of prime ideals of group algebras of Abelian torsion-free groups of finite rank over a field of positive characteristic

In the present paper certain methods are developed that enable one to study the properties of the controller of a prime faithful ideal of the group algebra of an Abelian torsion-free group of finite rank over a field . The main idea is that the quotient ring by the given ideal can be embedded as an integral domain into some field and the group becomes a subgroup of the multiplicative group of the field . This allows one to apply certain results of field theory, such as Kummer's theory and the properties of the multiplicative groups of fields, to the study of the integral domain . In turn, the properties of the integral domain depend essentially on the properties of the ideal . In particular, by using these methods, an independent proof of the new version of Brookes's theorem on the controllers of prime ideals of the group algebra of an Abelian torsion-free group of finite rank is obtained in the case where the field has positive characteristic.

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the moduleE(G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if RM is an arbitrary unitary left module and M+ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M+) under the canonical ring homomorphism R → E(M+). Then it holds that if E(M+)N is a pure submodule in E(M+)M+, then RN is a pure submodule in RM. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

Abelian Groups and Regular Modules

The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.

Duality and self-reflexive torsion-free abelian groups

We characterize the torsion-free abelian groups G of finite rank such that Hom(−, G) defines a rank preserving duality on the category of End( G)-submodules of finite sums of copies of G. Our results provide a maximal extension of Warfield's duality result for torsion-free groups of finite rank. In order to obtain our extension, we study the torsion-free abelian groups G for which G ≅ nat Hom(Hom( G, G), G).