Note on quasi-decompositions of irreducible groups'

TLDR: A quasi-decomposition theorem for a torsion free abelian group with quasi-endomorphism algebra satis- fying the minimum condition on left ideals was given in this paper.

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