Characterization of a class of torsion free groups in terms of endomorphisms

Abstract

!• Introduction and preliminaries* All groups considered here are subgroups of a fixed vector space V over the rational number field Q; we shall refer to these torsion free abelian groups simply as G will always denote a full subgroup of V, i.e., one with torsion quotient V/G. V is thus the divisible hull of G and r(G) = r(V), where r denotes rank. L(V) denotes the algebra of linear transformations of V. E{G) is the endomorphism ring of G and F(G) is the pure ideal of E(G) consisting of all endomorphisms with finite rank. Similarly, QE(G) is the quasi-endomorphism algebra of G and QF(G) is the ideal of elements having finite rank. Familiarity with the concept of quasi-isomorp hism is assumed; a complete background may be obtained from [2, 3, 9, 10]. g, —, = denote quasicontained, quasi-equal, and quasi-isomorphic, respectively. We consider QE(G) = {feL(V):fG QG}. Since each element of E(G) induces a unique linear transformation on V, we regard E(G) C QE(G) and use the same symbol to denote an endomorphism of G and also its induced linear transformation. All sums of groups are direct; e.g., notation such as G == A + B implies that A and B are disjoint groups. We take the following perspective. For feQE(G), define the final rank of / to be the minimum among the cardinal numbers r(fnG), n = 1, 2, . We assert the