Pure Subgroups Research Articles

Solution of a Problem of L. Fuchs Concerning Finite Intersections of Pure Subgroups

L. Fuchs states in his book “Infinite Abelian Groups” [6, Vol. I, p. 134] the followingProblem 13. Find conditions on a subgroup of A to be the intersection of a finite number of pure (p-pure) subgroups of A.The answer to this problem will be given as a special case of our theorem below. In order to find a better setting of this problem recall that a subgroup S ⊆ E is p-pure if pnE ∩ S = pnS for all natural numbers. Then S is pure in E if S is p-pure for all primes p. This generalizes to pσ-isotype, a definition due to L. J. Kulikov, cf. [6, Vol. II, p. 75] and [11, pp. 61, 62]. If α is an ordinal, then S is pσ-isotype if

Strongly Pure Subgroups of Separable Torsion-Free Abelian Groups

In this paper we prove that countable strongly pure subgroups of completely decomposable groups are completely decomposable. We also show that strongly pure subgroups of separable torsion-free groups are separable.

Strongly pure subgroups of separable torsion-free abelian groups

In this paper we prove that countable strongly pure subgroups of completely decomposable groups are completely decomposable. We also show that strongly pure subgroups of separable torsion-free groups are separable. One of the questions that have been considered concerning completely decom- posable groups is the following: under what conditions is a subgroup of a com- pletely decomposable again completely decomposable? The well-known Baer- Kaplansky-Kulikov theorem asserts that direct summands of completely decompos- able groups are also completely decomposable. It is well known that pure subgroups of completely decomposable groups are not necessarily completely decomposable and, in 1972, L. Bican characterized all those completely decomposable groups any pure subgroup of which is completely decomposable. Recently, we have shown that homogeneous pure subgroups of a completely decomposable G are completely decomposable provided that the extractible typeset of G is countable. In this note we partially extend the Baer-Kaplansky-Kulikov theorem to strongly pure subgroups of completely decomposable groups. Specifically, we show that in a completely decomposable whose typeset satisfies the maximum condition all strongly pure subgroups are completely decomposable. We also show that countable strongly pure subgroups of completely decomposable groups are completely decom- posable. This not only extends but also gives a new proof of the classical theorem of Kulikov on countable direct summands of completely decomposable groups. Fi- nally, we shall show that strongly pure subgroups of separable torsion-free groups are separable, extending the well-known theorem of L. Fuchs on direct summands of separable groups. Throughout this note the word will mean an abelian group and for notation and terminology the standard reference is volume II of Fuchs's book (2). Let G be torsion-free and completely decomposable, i.e. G is a direct sum of rank one torsion-free groups. A type r is said to be an extractible type oi G it G has a rank one summand of type r. The extractible typeset of G, denoted by £(G), is the set of all extractible types of G. For every r € £{G), we have G(t) — GT © G*(t), where GT is a nonzero homogeneous completely decomposable of type r; GT is called a maximal t-homogeneous summand of G. If G = @t<=£(g) Gt is a decomposition of G such that GT is a maximal r-homogeneous summand for every t G £(G), then this decomposition is known as a homogeneous decomposition of G.

Intersections of pure subgroups of valuated abelian groups

Intersections of pure subgroups of valuated abelian groups

SOME CONDITIONS FOR EMBEDDABILITY OF AN $ FC$-GROUP IN A DIRECT PRODUCT OF FINITE GROUPS AND A TORSIONFREE ABELIAN GROUP

By definition, a torsionfree abelian group belongs to the class if every -group with and is embeddable in a direct product of finite groups and a torsionfree abelian group.If is a torsionfree abelian group of rank 1, then a prime .The fundamental result of the article is the following statement. Theorem. A torsionfree abelian group belongs to the class if and only if it admits a series of pure subgroups with the following properties:(I) The quotient is of rank 1, and the set is finite, .(II) For every prime , there exists a number such that whenever .Bibliography: 9 titles.

Intersections of Pure Subgroups in Abelian Groups

It is shown that a subgroup $H$ of an abelian group $G$ is an intersection of pure subgroups of $G$ if and only if, for all primes $p$ and positive integers $n$, ${p^n}g \in H$ and ${p^{n - 1}}g \notin H$ imply that there exists $z \in G$ such that ${p^n}z = 0$ and ${p^{n - 1}}z \notin H$. This solves a problem posed by $L$. Fuchs in [2] and [3].

The elementary theory of Abelian groups with m-chains of pure subgroups

The elementary theory of Abelian groups with m-chains of pure subgroups

Intersections of pure subgroups in abelian groups

It is shown that a subgroup H H of an abelian group G G is an intersection of pure subgroups of G G if and only if, for all primes p p and positive integers n n , p n g ∈ H {p^n}g \in H and p n − 1 g ∉ H {p^{n - 1}}g \notin H imply that there exists z ∈ G z \in G such that p n z = 0 {p^n}z = 0 and p n − 1 z ∉ H {p^{n - 1}}z \notin H . This solves a problem posed by L L . Fuchs in [2] and [3].

A theorem on free envelopes

The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665-682] to be a finitely generated free commutative semigroup F ( S ) F(S) with identity and a homomorphism α : S → F ( S ) \alpha :\,S\, \to \,F(S) endowed with certain properties. Grillet raised the following question: does α ( S ) \alpha (S) always generate a pure subgroup of the free Abelian group with the same basis as F ( S ) F(S) ? We prove this is indeed the case. It follows as a result of two lemmas. Lemma 1: Given a full rank proper subgroup H of a finitely generated free Abelian group F and a basis X of F there exists a surjective homomorphism f : F → Z f:\,F\, \to \,{\textbf {Z}} such that f is positive on X and f | H {f_{\left | H \right .}} is not surjective. Lemma 2: A finitely generated totally cancellative reduced subsemigroup of a finitely generated free Abelian group F is contained in the positive cone of some basis of F. The following duality theorem is also proved. Let S ∗ ≅ Hom ⁡ ( S , N ) {S^{\ast }}\, \cong \,\operatorname {Hom} (S,\,N) where N is the nonnegative integers under addition. Then S ≅ S ∗ ∗ S\, \cong \,{S^{{\ast }{\ast }}} if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.

Quasi-P-Pure-Injective Groups

Recently, a great deal of attention has been paid to the concept of quasipure injectivity introduced by L. Fuchs as Problem 17 in [5]. An abelian group G is said to be quasi-pure-injective (q.p.i.) if every homomorphism from a pure subgroup of G to G can be lifted to an endomorphism of G. D. M. Arnold, B. O'Brien and J. D. Reid have succeeded in [1] to characterize torsion free q.p.i. of finite rank, whereas in [2] we solved the torsion case and in [3] we studied certain classes of infinite rank torsion free q.p.i. groups.

Pure Subfields of Purely Inseparable Field Extensions

The notion of pure subgroups is due to Prufer [7]. It has proven extremely useful in establishing structural properties of abelian groups. In a recent paper [9], Waterhouse introduced the concept of a pure subfield of a purely inseparable extension. Let L be a purely inseparable modular extension of k, and let K be an intermediate field. K is called pure if K and k(Lpn) are linearly disjoint over k(Kpn) for all n. Waterhouse used this concept to establish the existence of basic subfields [9].

INDUCTIVE PURITIES IN ABELIAN GROUPS

In the paper we study purities in categories of Abelian groups having the property that the union of an increasing chain of -pure subgroups of an Abelian group is itself an -pure subgroup of . Such purities are called inductive. For every prime number we set if for , , there is an and an such that . Head purities are defined as purities of the form , where is a set of prime numbers. Head purities and -purities, evidently, are inductive. In the paper we show that every inductive purity in the category of all torsion-free Abelian groups is a certain -servancy, every inductive purity in the category of all periodic Abelian groups is a certain -purity, and every inductive purity in the category of all Abelian groups is the intersection of a certain -purity and a certain Head purity.Bibliography: 8 items.

On Pure Subgroups of LCA Groups

On Pure Subgroups of LCA Groups

Pure N-high Subgroups, P-adic Topology and Direct Sums of Cyclic Groups

This paper is divided into two sections. In the first, we characterize the subgroups N of a reduced abelian primary group for which all pure N-high subgroups are bounded. This condition on pure N-high subgroups occurs in several instances, for instance, all pure N-high subgroups of a primary group G are bounded if G is the smallest pure subgroup of G containing N; all N-high subgroups are bounded if N ≠ 0 and all N-high subgroups are closed in the p-adic topology.

Conjugately Pure Subgroup Problems

Conjugately Pure Subgroup Problems

Conjugately Pure Subgroup Problems

Conjugately Pure Subgroup Problems

Completely decomposable abelian groups any pure subgroup of which is completely decomposable

Completely decomposable abelian groups any pure subgroup of which is completely decomposable

A Class of Pure Subgroups of Completely Decomposable Abelian Groups

A Class of Pure Subgroups of Completely Decomposable Abelian Groups

Abelian Groups in Which Every α-Pure Subgroup is β-Pure

The determination of the abelian groups in which every neat subgroup is pure is a relatively routine exercise (see [6]). There are numerous problems of this type; for example, the determination of the groups in which every pure subgroup is isotype or the groups in which every subgroup is isotype. These are all special cases of the general problem of determining the abelian groups in which every α-pure subgroup is β-pure for arbitrary ordinal numbers α and β. The solution of this general problem is the object of this paper.

Direct Products and Sums of Torsion-Free Abelian Groups

Let $A$ be a finite rank, indecomposable torsion-free Abelian group whose $p$-ranks are less than two for all primes $p$. Let $G$ be a direct product of copies of $A$, and $B$ be a nonzero countable pure subgroup of $G$ such that $B$ is the span of the homomorphic images of $A$ in $B$. Then it is shown that $B$ is a direct sum of copies of $A$. This result is applied to obtain a Krull-Schmidt theorem for direct sums of groups $A$ from a semirigid class of groups. In particular, if the groups $A$ have rank one, then the well-known BaerKulikov-Kaplansky theorem is obtained.