Vol. 51, No. 1, 1974

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Maximal pure subgroups of torsion complete abelian p-groups

John St. Clair Werth, Jr.

Vol. 51 (1974), No. 1, 307–316

Let N be the set of nonnegative integers, and let B = Σ [bi](i N) be the direct sum of cyclic groups with 0(bt) = pi+1. Denote by B the torsion-completion of B. This paper is concerned with pure subgroups of the group B. If G is such a group, let

I(G ) = {i|i- th Ulm invariant of G is nonzero}.

Beaumont and Pierce introduced a further invariant for G, namely,

U(G) = {I(A )|A is a pure torsion- complete subgroup of G}.

U(G) is a (boolean) ideal in 𝒫(N), the power set of N.

If is an ideal in 𝒫(N), lhen the canonical example of a pure subgroup, G, of B with U(G) = is constructed as follows:

G = 𝒢(ℐ) = ΣAI(I ∈ ℐ ) where AI is the torsion-completion of Σ ⊕ [bi](i ∈ I).

Beaumont and Pierce showed that if 𝒫(N)has no atoms and is free, then there exist maximal pure subgroups G of B such that G ⊃𝒢() and U(G) = . The purpose of this paper is to give necessary and sufficient conditions for the existence of such a G in the case that 𝒫(N)is finite. In the process, some information is obtained about the number of nonisomorphic extensions of 𝒢().

Mathematical Subject Classification 2000
Primary: 20K10
Received: 26 September 1972
Published: 1 March 1974
John St. Clair Werth, Jr.