MODULES OVER PIDs THAT ARE INJECTIVE OVER THEIR ENDOMORPHISM RINGS

Abstract

This chapter describes the modules over PIDs that are injective over their endomorphism rings. An abelian group is called algebraically compact if it is pure injective as a Z-module. This is one of the few tractable classes of abelian groups. The chapter presents a complete structure theory. It is found that an injective module over any ring E is algebraically compact as an abelian group. The additive group of any pure—injective E—module is algebraically compact. The question of deciding as to which (algebraically compact) groups can be additive groups of injective modules remains open, but those abelian groups G that are injective when viewed as modules over their endomorphism rings E(G) are characterized. It is assumed that R is a principal ideal domain and K its quotient field. It is observed that if p ∈ R is a prime, the completion of R in the p-adic topology is denoted by R, its quotient field is denoted by K, and the rank-one divisible torsion Rp-module, Kp/Rp, is denoted by Rp∞.