Title: Endomorphism rings and subgroups of finite rank torsion-free Abelian groups
Abstract: A be a finite rank torsion-free abelian group and let E(A) denote the endomorphism ring of A. Then Q ® z E(A) = QE(A) and E(A)/pE(A) are artinian algebras, where Z is the ring of integers, Q is the field of rational, and p is a prime of Z.Define A to be Q-simple if QE(A) is a simple algebra, and p-simple for a prime p of Z if /?£(/4) = £(y4) or if E(A)/pE(A) is a simple algebra.In contrast to finite rank torsion-free groups in general, groups that are psimple for each p have some pleasant decomposition properties. THEOREM I. A reduced group A is p-simple for each prime p of Z if and only ifand p-simple for each prime p ofZ, and ifp is a prime of Z then there is somej with AjpA = AjjpAj. THEOREM II. A group A is Q-simple and p-simple for each prime p of Z if and only ifwhere each B { is strongly indecomposable, Q-simple and p-simple for each prime p of Z and B { is nearly isomorphic to Bj (in the sense of Lady [7]) for each i andj.Suppose that A is g-simple and /7-simple for each prime p of Z. Then A is indecomposable if and only if A is strongly indecomposable.Furthermore, if S = Center E(A), then S is a subring of an algebraic number field such that every element of S is a rational integral multiple of a unit of S, as described in [1], and E(A) is a maximal S-order in QE(A).Examples of groups that are Q-simple and /7-simple for each prime p of Z include: indecomposable strongly homogeneous groups (characterized in [1]); indecomposable groups with /?-rank g 1 for each prime p of Z (Murley [8]); and indecomposable quasi-pure-projective and quasi-pureinjective groups ([4]).Define A to be irreducible if QA is an irreducible QE(A)-module (Reid [10]) and p-irreducible, for a prime p of Z, if AjpA is an irreducible E(A)I /?iì(y4)-module.If A is irreducible (/?-irreducible), then A is g-simple •Research supported, in part, by N.S.