Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

Almost Isomorphic Torsion-Free Abelian Groups and Similarity of Homogeneously Decomposable Groups

In the paper the concept of the similarity of homogeneously decomposable groups is introduced. One investigates the cases in which almost isomorphic groups are similar or isomorphic. Mathematics Subject Classification (2000) 20K99.     

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.     

Self-Cancellation of Torsion-Free Abelian Groups of Finite Rank

A very simple example of an Abelian group free of self-cancellation is constructed. Bibliography: 11 titles.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Mathematical Notes - Let C be an Abelian group. A class X of Abelian groups is called a CE?H-class if, for every groups A, B ∈ X, the isomorphisms E?(A) ? E?(B) and...     

A Class of Strongly Decomposable Abelian Groups

Let G be a completely decomposable torsion-free Abelian group and G= G_i, where G _i is a rank 1 group. If there exists a strongly constructive numbering of G such that (G,) has a recursively enumerable sequence of elements g _i G _i , then G is called a strongly decomposable group. Let pi, i, be some sequence of primes whose denominators are degrees of a number p _i and let . A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that for some numbers i ₁,...,i _k . We bring in the concept of a quasihyperhyperimmune set, and specify a necessary and sufficient condition on the characteristic of A subject to which the group in question is strongly decomposable. Also, it is proved that every hyperhyperimmune set is quasihyperhyperimmune, the converse being not true.     

Notes on Essentially Finitely Indecomposable Nonthick Primary Abelian Groups

Let R be a commutative Noetherian ring of prime characteristic and M be an x-divisible right R[x, f]-module that is Noetherian as R-module. We give an affirmative answer to the question of Sharp and Yoshino in the case where R is semilocal and prove that the set of graded annihilators of R[x, f]-homomorphic images of M is finite. We also give a counterexample in the general case.     

Representations of Knot Groups and Twisted Alexander Polynomials

We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.     

Induced Representations of Abelian Groups of Finite Rank

We prove that any irreducible faithful representation of an almost torsion-free Abelian group G of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group G.     

换位子群是不可分Abel群的有限秩可除幂零群

完整地确定了换位子群是不可分Abel群的有限秩可除幂零群的结构,证明了下面的定理.设G是有限秩的可除幂零群,则G的换位子群是不可分Abel群当且仅当G'=Q或Q_p/Z且G可以分解为G=S×D,其中当G'=Q时,■当G'=Q_p/Z时,S有中心积分解S=S_1*S_2*…*S_r,并且可以将S形式化地写成■其中■,式中s,t都是非负整数,Q是有理数加群,π_κ(k=1,2,…,t)是某些素数的集合,满足π_1■Cπ_2■…■π_t,Q_π_k={m/n|(m,n)=1,m∈Z,n为正的π_k-数}.进一步地,当G'=Q时,(r;s;π_1,π_2,…,π_t)是群G的同构不变量;当G'=Q_p/Z时,(p,r;s;π_1,π_2,…,πt)是群G的同构不变量.即若群H也是有限秩的可除幂零群,它的换位子群是不可分Abel群,那么G同构于H的充分必要条件是它们有相同的不变量.     

On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings

Mathematical Notes - Let $$C$$ be an Abelian group. A class $$X$$ of Abelian groups is called a $$_CH $$ -class (a $$_CEH$$ -class) if, for any groups $$A$$ and $$B$$ in the class $$X$$ , the...     

关于Banach空间的遗传不能分解和商遗传不能合成性质

该文说明遗传不能分解空间上的黎斯算子类构成了最大、非平凡的算子理想，简化了Gowers和Maurey关于这类空间上算子构成的推证．应用对偶原理，引入商遗传不能合成的概念，讨论商遗传不能合成的Banach空间上的算子构成，得到了相应的一些结果．     

$${\text{IA}}$$ -Automorphisms of Free Products of Two Abelian Torsion-Free Groups

Let be the free product of two Abelian torsion-free groups, let and , where is the Cartesian subgroup of the group , and let F contain no zero divisors. In the paper it is proved that, in this case, any automorphism of the group is inner. This result generalized the well-known result of Bachmuth, Formanek, and Mochizuki on the automorphisms of groups of the form , , , where is a free group of rank two.     

Indecomposable Modules and Gelfand Rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R _P is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring.     

Galois Stability, Integrality and Realization Fields for Representations of Finite Abelian Groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups GGL _n (E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices gG to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.     

Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

Compact Operators on Homogeneous Banach Spaces of Distributions on Locally Compact Abelian Groups

In this paper, two theorems about the compactness of almost invariant operators on homogeneous Banach spaces of distributions (in the sense of Feichtinger [11]) defined on a locally compact abelian group are proved. Our theorems generalize the corresponding results of K. de Leeuw [3] and Tewari and Madan [18] for operators on homogeneous Banach spaces on the circle group and Segal algebras on a compact abelian groups, respectively.AMS Subject Classification 2000: 43A85, 47B07.