Decomposable Torsion-free Abelian Groups Research Articles

Enumerations and Completely Decomposable Torsion-Free Abelian Groups

Enumerations and Completely Decomposable Torsion-Free Abelian Groups

Torsion-free groups and modules with the involution property

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of p -adic integers.

The Jacobson Radical of an Endomorphism Ring for an Abelian Group

The Jacobson radical of an endomorphism ring is computed for a completely decomposable torsion-free Abelian group and for a mixed Abelian group in one class of mixed groups. For the latter case, we also look into the question when a factor ring w.r.t. the radical is regular in the sense of Nuemann.

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 \(\mathcal{X}\) of Abelian groups is said to be CH-definable in the class \(\mathcal{X}\) if, for any group B\in\(\mathcal{X}\), 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 \(\mathcal{X}\) is CH-definable in \(\mathcal{X}\), then the class \(\mathcal{X}\) is called an CH-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a CH-class, where C is a completely decomposable torsion-free Abelian group.

Computation of the Mapping of Factorization by the Radical for K 0 + of the Endomorphism Ring

The mapping of factorization by the radical is computed for the semigroup of projective, finitely generated modules over the endomorphism ring of an almost completely decomposable torsion-free Abelian group of finite rank that is divisible by almost all prime numbers. Also, an answer is given to the question concerning the collections of groups of rank 1 for which one can construct an almost completely decomposable group, indecomposable as an object in \(\bar M^p\), by adding a generator. Bibliography: 4 titles.

A Class of Strongly Decomposable Abelian Groups

Let G be a completely decomposable torsion-free Abelian group and G= ⊕ Gi, where Gi is a rank 1 group. If there exists a strongly constructive numbering ν of G such that (G,ν) has a recursively enumerable sequence of elements gi ∈ Gi, then G is called a strongly decomposable group. Let pi, i∈ω, be some sequence of primes whose denominators are degrees of a number pi and let \(\mathop \oplus \limits_{i \in \omega } Q_{Pi} \). A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that \(p_{i_1 } = ... = p_{i_k } = p\) for some numbers i1,...,ik. 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.

STACKED BASES FOR COUNTABLE HOMOGENEOUS COMPLETELY DECOMPOSABLE GROUPS

Let A ⊆ X be two homogeneous completely decomposable torsion free abelian groups of the same type τ and of countable rank such that the quotient X/A is a direct sum of torsion cyclic groups and a homogeneous completely decomposable summand of type τ. Furthermore assume that A and X have a common direct summand of countable rank. We show that there exist stacked bases for A and X, i.e. there exist xi ∈ X (i ∈ ω) and di ∈ Z (i ∈ ω) such that and . This proves a stacked bases theorem for pairs of homogeneous completely decomposable torsion free abelian groups of countable rank and the same type with a large commmon summand.

Separable pure subgroups of completely decomposable torsion-free abelian groups

Separable pure subgroups of completely decomposable torsion-free abelian groups

On automorphisms of completely decomposable torsionfree abelian groups

On automorphisms of completely decomposable torsionfree abelian groups

An Extension of the Theory of Completely Decomposable Torsion-Free Abelian Groups

We construct a class of strongly indecomposable finite rank torsion-free groups that includes the rank-one groups, and develop a complete set of invariants for them and their direct sums.

An extension of the theory of completely decomposable torsion-free abelian groups

We construct a class of strongly indecomposable finite rank torsion-free groups that includes the rank-one groups, and develop a complete set of invariants for them and their direct sums.

Prebasic submodules over valuation rings

Basic concepts in the theory of modules over valuation rings are introduced. The notion of height is used to define indicators of elements, whose irregularities are investigated. The indicator leads to the new notion of smoothness, a property which does not originate from abelian groups. Invariants generalizing the finite Ulm-Kaplansky invariants of abelian p-groups, as well as the Baer invariants for completely decomposable torsion-free abelian groups, are defined, and several results relating these invariants of a module to those of submodules are proved. All these concepts lead to the notion of prebasic submodules, which seems to be the right analogue of the basic subgroups in abelian groups.

Almost Completely Decomposable Torsion Free Abelian Groups

A finite rank torsion free abelian group $G$ is almost completely decomposable if there exists a completely decomposable subgroup $C$ with finite index in $G$. The minimum of $[G:C]$ over all completely decomposable subgroups $C$ of $G$ is denoted by $i(G)$. An almost completely decomposable group $G$ has, up to isomorphism, only finitely many summands. If $i(G)$ is a prime power, then the rank 1 summands in any decomposition of $G$ as a direct sum of indecomposable groups are uniquely determined. If $G$ and $H$ are almost completely decomposable groups, then the following statements are equivalent: (i) $G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group $L$. (ii) $i(G) = i(H)$ and $H$ contains a subgroup $G’$ isomorphic to $G$ such that $[H:G’]$ is finite and prime to $i(G)$. (iii) $G \oplus L \approx H \oplus L$ where $L$ is isomorphic to a completely decomposable subgroup with finite index in $G$.

Almost completely decomposable torsion free abelian groups

A finite rank torsion free abelian group G G is almost completely decomposable if there exists a completely decomposable subgroup C C with finite index in G G . The minimum of [ G : C ] [G:C] over all completely decomposable subgroups C C of G G is denoted by i ( G ) i(G) . An almost completely decomposable group G G has, up to isomorphism, only finitely many summands. If i ( G ) i(G) is a prime power, then the rank 1 summands in any decomposition of G G as a direct sum of indecomposable groups are uniquely determined. If G G and H H are almost completely decomposable groups, then the following statements are equivalent: (i) G ⊕ L ≈ H ⊕ L G \oplus L \approx H \oplus L for some finite rank torsion free abelian group L L . (ii) i ( G ) = i ( H ) i(G) = i(H) and H H contains a subgroup G ′ G’ isomorphic to G G such that [ H : G ′ ] [H:G’] is finite and prime to i ( G ) i(G) . (iii) G ⊕ L ≈ H ⊕ L G \oplus L \approx H \oplus L where L L is isomorphic to a completely decomposable subgroup with finite index in G G .

A class of pure subgroups of completely decomposable abelian groups

The class M of pure subgroups of completely decomposable torsion free abelian groups of finite rank is considered by Butler [2]. Many of the examples of pathological direct sum decompositions of finite rank torsion free abelian groups occur in the class .4. On the other hand, homogeneous a-groups are completely decomposable and if A is an s-group, then typeset(A) is finite and closed under intersection of types. This note considers a-groups with typesets of cardinality at most 4. Motivation is provided by the curious results of Cruddis [3]. An easier and more illuminating proof of these results is a consequence of the following theorems. Let A= {-To, TI, T2, T73} be a set of types with To =rl1r -72r lT3; let .A be the class of a-groups with typeset c A; let :A be the sum of types in A; and let >,A be the sum of types in typeset(A). Define T to be the type represented by (Xo) and, forp a prime, let T-D be the type represented by (ne) where n2,=O and nQ= xo if p#q.

Isomorphism conditions for completely decomposable torsion-free abelian groups with isomorphic rings of endomorphisms

Assuming the generalized continuum hypothesis, we obtain necessary and sufficient conditions for completely decomposable torsion-free abelian groups with isomorphic rings of endomorphisms to be isomorphic.

Some properties of completely decomposable torsion free Abelian groups

Some properties of completely decomposable torsion free Abelian groups