Showing papers on "Torsion-free abelian group published in 1970"

Unit groups of infinite abelian extensions

Abstract: Let F be a finite extension field of the rational numbers, 0, and let K be an infinite abelian extension of F. Let 5 be a finite set of prime divisors of Q including the Archimedean one. An 5-unit of K is a field element which is a local unit at all prime divisors of F which do not restrict on Q to a member of 5. It is shown that the group of 5-units of K is the direct product of the group of roots of unity of K with a free abelian group. The question of determining the structure of the unit group in certain infinite extensions of Q arose in some work of the author [2] on group algebras. In that case, the field K is the maximal cyclotomic extension of F. The conclusion of the theorem that we prove has been obtained (for entirely different purposes) by DiBello [l] for a restricted class of abelian extensions, one not including the maximal cyclotomic ones. Since the result is somewhat unexpected, it seems worthwhile to give an exposition of it which is independent of the applications. We first prove a simple lemma on groups. If G is an abelian group and H a subgroup, we define the purification of H in G, denoted (H)x, to be the set of all xEG such that xnEH for some positive integer, n. In other words, (H)x is the inverse image of the torsion subgroup of G/H under the natural map. Let p be a prime number. We define the p-purification of H in G, denoted (H)P, to be the set of all x£G such that xp'EH for some positive integer, r. Then (H)p is the inverse image of the ^-torsion subgroup of G/H under the natural map. Lemma. Let G be a countably generated torsion free abelian group. Assume that for any finitely generated subgroup, H', there exists a subgroup, H, containing H', such that the purification of H in G is finitely generated. Then G is free. Proof. Let gi, g2, • • ■ be generators for G. By induction we shall construct finitely generated subgroups, {IP,l^i}, and subsets, {Bi11 Si}, of G such that for any positive integer, n, we have that gi, • ■ • . gn are in Wn, G/Wn is torsion free, Bn is a free basis for Wn, and 5iCB2C • • C5„. If this can be done, then G = U,Tr7,implies that U,Bi is a free basis for G. Received by the editors November 4, 1969. AMS Subject Classifications. Primary 1065, 1066, 1250; Secondary 2080.

Note on quasi-decompositions of irreducible groups'

Abstract: A quasi-decomposition theorem is obtained for a torsion free abelian group with quasi-endomorphism algebra satis- fying the minimum condition on left ideals. Several quasi-decomposition theorems of J. D. Reid and R. S. Pierce for torsion free abelian groups of finite rank can be extended to groups of arbitrary rank by replacing the finite rank hypothesis with the requirement that the quasi-endomorphism algebra of the groups satisfy the minimum condition on left ideals. This minimum condi- tion can also be characterized topologically. The lemma below is the key to these generalizations; that they are nontrivial is illustrated by familiar examples such as groups of p-adic integers. Hereafter the term "group" refers to a reduced, torsion free abelian group. The discussion is normalized by considering only subgroups of a fixed vector space V over the rational number field Q. The algebra of linear transformations of V, L(V), is equipped with the finite topology (4). Basic results about quasi-isomorphism are assumed; for a complete background consult (1), (6), (7). C, -, denote quasi-contained, quasi-equal, and quasi-isomorphic, respectively. G will always denote a full subgroup of V, i.e., a subgroup with torsion quotient, V/G. Recall that QE(G) = {fCL(V):fGCG } is the quasi- endomorphism algebra of G. H* denotes the rational subspace of V spanned by a subgroup H of V. Finally, all sums are direct. After Reid (7), call G irreducible if and only if it has no nontrivial pure, fully invariant subgroups. It is easy to see that an irreducible group is homogeneous; thus we lose no generality by considering only reduced groups. Reid shows that G is irreducible if and only if V is an irreducible QE(G)-module by establishing a one-to-one correspon- dence between the pure, fully invariant subgroups of G and the QE(G)-submodules of V. It follows readily that a quasi-summand of an irreducible group is itself irreducible and thus that irreducibility