Detachablep-groups and quasi-injectivity

Abstract

If E is a ring, and G is an E-module, then G is quasi-injective if every homomorphism from a submodule of G to G can be extended to an endomorphism of G. POOLE and REID [2] raise the question as to which abelian groups G are quasiinjective as modules over their endomorphism rings E. They prove that direct sums of cyclic p-groups are quasi-injective. This result is extended here to a large class of p-groups, including all p-groups that have no elements of infinite height, and all totally projective p-groups -- in particular, all countable p-groups. The central idea is a property shared by totally projective p-groups and pgroups with no elements of infinite height. DEFINITION. If G is a p-group and xEG[p], then x is said to be detachable if G can be written as G1 @ Gz with (x) =p'G~ [p] for some ordinal c~. If every element of G[p] is detachable, we say that G is detachable. We write p'G~[p] instead of p~G~ to allow Ga to be divisible. Note that every element in G[p] of finite height is detachable, so every p-group with no elements of infinite height is detachable. Hill's extension of Ulm's theorem to totally projective p-groups (see [3]) implies that these groups are also detachable. THEOREM 1. If G is a detachable p-group, and x, yEG[p], then there is an endomorphism f of G such that f(x) =y or f(y) = x.