Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.     

When are all group codes of a noncommutative group Abelian (a computational approach)?

Let G be a finite group and F be a field. Any linear code over F that is permutation equivalent to some code defined by an ideal of the group ring FG will be called a G-code. The theory of these ??abstract?? group codes was developed in 2009. A code is called Abelian if it is an A-code for some Abelian group A. Some conditions were given that all G-codes for some group G are Abelian but no examples of non-Abelian group codes were known at that time. We use a computer algebra system GAP to show that all G-codes over any field are Abelian if |G|?<?128 and |G| ? {24, 48, 54, 60, 64, 72, 96, 108, 120}, but for F?=? $ {\mathbb{F}_5} $ and G?=?S₄ there exist non-Abelian G-codes over F. It is also shown that the existence of left non-Abelian group codes for a given group depends in general on the field of coefficients, while for (two-sided) group codes the corresponding question remains open.     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

Absolute Nil-Ideals of Abelian Groups

It is known that in an Abelian group G that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on G is $\bigcap\limits_{p} pT(G)$ , where T(G) is the torsion part of G. In this work, we define a pure fully invariant subgroup G*???T(G) of an arbitrary Abelian mixed group G and prove that if G contains no nonzero torsion-free subgroups, then the subgroup $\bigcap\limits_{p} pG^{*}$ is a nil-ideal in any ring on G, and the first Ulm subgroup G1 is its nilpotent ideal.     

Local spectral synthesis on Abelian groups

We say that a complex valued function defined on an Abelian group G is a local polynomial, if its restriction to every finitely generated subgroup of G is a polynomial. We prove that local spectral synthesis (that is, spectral synthesis using local polynomials instead of polynomials) holds on every Abelian group having countable torsion free rank. More precisely, there is a cardinal ω ₁≦κ≦2 ^ω such that local spectral synthesis holds on an Abelian group G if and only if the torsion free rank of G is less than κ.     

On Chow Groups of G-Graded Rings

Abstract

Let A be a Noetherian ring graded by a finitely generated Abelian group G. It is shown that a Chow group A _?(A) of A is determined by cycles and a rational equivalence with respect to certain G-graded ideals of A. In particular, A _?(A) is isomorphic to the equivariant Chow group of A if G is torsion free.     

Primary decomposition over rings graded by finitely generated Abelian groups

Let A be a Noetherian ring which is graded by a finitely generated Abelian group G. In general, for G-graded modules there do not exist primary decompositions which are graded themselves. This is quite different from the case of gradings by torsion free group, for which graded primary decompositions always exists. In this paper we introduce G-primary decompositions as a natural analogue to primary decomposition for G-graded A-modules. We show the existence of G-primary decomposition and give a few characterizations analogous to Bourbaki's treatment for torsion free groups.     

Discriminating Completions of Hyperbolic Groups

A group G is called an A-group, where A is a given Abelian group, if it comes equipped with an action of A on G which mimics the way in which Z acts on any group. This action is codified in terms of certain axioms, all but one of which were introduced some years ago by R. C. Lyndon. For every such G and A there exists an A-exponential group G ^A which is the A-completion of G. We prove here that if G is a torsion-free hyperbolic group and if A is a torsion-free Abelian group, then the Lyndon's type completion G ^A of G is G-discriminated by G. This implies various model-theoretic and algorithmic results about G ^A .     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

Bounded sets in topological groups and embeddings

We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G?(K?A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups.     

On Quasi-Regular Torsion-Free Modules

A torsion-free module is called quasi-regular if each cyclic submodule is a quasi-summand. This article characterizes torsion-free Abelian groups that are quasi-regular as modules over a subring of their endomorphism ring. In particular, if G is a torsion-free Abelian group such that its ring Q E of quasi-endomorphisms is Artinian, then the left E-module G is quasi-regular if and only if the left C-module G is quasi-regular, where C is the center of its endomorphism ring E.     

Torsion-free modules with UA-rings of endomorphisms

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring End _R (A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.     

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Torsion units in integral group rings of some metabelian groups,II

We prove that any torsion unit of the integral group ring ZG is rationally conjugate to a trivial unit if G = A ⋊ X with both A and X abelian, |Xz.sfnc; < p for every prime p dividing |A| provided either |X| is prime or A ic cyclic.     

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ? J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.     

Nonreduced Abelian groups with UA-rings of endomorphisms

A ring K is a unique addition ring (a UA-ring) if its multiplicative semigroup (K, · ) can be equipped with a unique binary operation + transforming this semigroup to a ring (K, ·, +). An Abelian group is called an End-UA-group if its endomorphism ring is a UA-ring. In the paper, we find End-UA-groups in the class of nonreduced Abelian groups.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.     

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 module _E (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 _R M 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 _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.