On endomorphism rings of non-separable abelian p-groups

Abstract

There have been spectacular advances in recent years in the so-called realization problem for certain classes of abelian groups and modules. The advances have derived mainly from powerful combinatorial set-theoretic tools pioneered by Shelah. In the case of separable abelian p-groups these results appear, e.g., in [ 1,4, 51. For non-separable abelian p-groups the most significant contribution has been [2]. Our objective in this paper is to extend the results in [2] and derive analogous results to those obtained in [4, 51. It is perhaps worth pointing out that our methods are entirely algebraic; the necessary set-theoretical work has been carried out in the separable case and no further set-theoretical arguments are needed. In this the work is reminiscent of [S]. All our set theory (with one exception) is in ZFC and this is not acknowledged in the statement of individual results. In the exceptional case where we work in Godel’s Constructible Universe (V= L) this is acknowledged by appending (V= L) after the statement number. The principal realization result can be stated as: