A NOTE ON THE COUNTABLE EXTENSIONS OF SEPARABLE p–PROJECTIVE ABELIAN p–GROUPS

Throughout this brief note all groups are assumed to be abelian p-primary, written additively as is customary when regarding the group structure. Since we shall deal exclusively only with p-torsion abelian groups, for some arbitrary but a fixed prime p, there should be no confusion in future removing the phrase ”is an abelian p-group”. Concerning the terminology, under the term a separable group, we mean a reduced group without elements of infinite height (as computed in the full group). All other notation and notions are standard. For example, for any group A, the letter A = pA = ∩k<ωp A traditionally denotes the first Ulm subgroup of A. Before stating and proving our main attainment, to make the article more nearly self contained, we give a systematic introducing in the basic concepts of the principle best known results in this theme. In [5], Wallace has proven the following important assertion, which he used in the classification of rank one mixed abelian groups having totally projective primary components.