Ladislav Bican

Butler groups of infinite rank

The class of pure subgroups of completely decomposable groups of finite rank was introduced and investigated by Butler in [5]. Lady called the groups in this class “Butler groups”. Some other authors studied Butler groups under different names (see Koehler [9] and the first author [3] and [4]). Recently, Arnold collected the known results on Butler groups in [1], and investigated more deeply this class of finite rank groups.

Subgroups of Butler Groups

All groups in question are abelian. If x is an element of a torsionfree group G then |x| G , or simply |x| is the characteristic and t G (x) = t(x) is the type of x in G. Each torsionfree group G can be embedded essentially in a Q-vector space Q ⊗ G (= the divisible hull). The Q-dimension of this vector space is called the rank of G. Especially, the rank one groups are just the subgroups of the additive group (Q, +) of all rational numbers, they are often called rational groups. A torsionfree group G is said to be completely decomposable, if it is (up to isomorphism) a direct sum of rational groups. The corank of a pure subgroup H of G is the rank of G/H. For unexplained terminology and notation see [15]. As usual, (CH) denotes the continuum hypothesis, i.e. \( {2^{{N_0}}} = {N_1}\). By a smooth (increasing) union of a group G we mean a collection of pure subgroups G α indexed by an initial segment of ordinals with the property that G β ≤ G α when β w G n of pure subgroups, where G n ≤ G n+1 for each n

Smooth unions of Butler groups

Abstract If is the union of a smooth strictly ascending chain of B 2-subgroups G α, then, in the case when κ < ℵω, a criterion is established under which Gbecomes a B 2-group. This criterion is dependent on a new class of torsion-free groups and generalizes earlier criteria for freeness established by Paul Hill. The result of S. Shelah and others establishing that, for a weakly compact cardinal κ “κ-free” implies “free”, is extended to the case of B 2-groups.

Pure Subgroups of Butler Groups

In [5]the class of Butler groups of arbitrary rank was introduced. Recently, in [6], it was shown that the class of Butler groups with ordered type set is closed under countable pure subgroups, while the general case remained open. In this note we shall show that the answer to the last questionis negative, while the class of Butler groups with inversely well-ordered type set is closed under arbitrary pure subgroups. Moreover, we extend [5;Proposition 3.5] to arbitrary homogeneous groups by showing that a homogeneous torsionfree group is Butler if and only if it is completely decomposable. The main tool in this direction is a slight modification of Griffith’s proof [9] of the freeness of Baer’s

Butler groups as smooth ascending unions

Fuchs and Rangaswamy [8] proved that a smooth ascending union G = U α<λ Gα of pure B2-subgroups is a B2-group provided either λ = ωo or λ = ω1,/G/ = N1 and all the subgroups Gα are decent in G. In this paper, we extend this result to arbitrary cardinal numbers λ under suitable hypotheses on the factor-groups G α+1/G α.

Lattice Properties of Torsion Theories

The main purpose of this brief note is to show that the sets of all hereditary torsion theories which are noetherian, strongly noetherian, or of finite type, respectively, form a sublattice of the lattice of all hereditary torsion theories for the category R-mod.

Relative purity over Noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality λ ≥ |R|, then its injective hull E(M) is of the same size. Further, if M is an injective module with |M| ≥ (2λ)+ and K ≤ M is its submodule such that |M/K| ≤ λ, then K contains an injective submodule L with |M/L| ≤ 2λ. These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author’s previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3–6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105–107], [Pure subgroups, Math. Bohem. 126 (2001), 649–652].

Almost Butler groups

Generalizing the notion of the almost free group we introduce almost Butler groups. An almost B2-group G of singular cardinality is a B2-group. Since almost B2-groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that G is a B1-group. Some other results characterizing B2-groups within the classes of almost B1-groups and almost B2-groups are obtained. A theorem of [BR] stating that a group G of weakly compact cardinality λ having a λ-filtration consisting of pure B2-subgroup is a B2-group appears as a corollary.