Slenderness, Completions, and Duality for Primary Abelian Groups Original Research Article

IfAis a fixed abelian group with endomorphism ringE, then for any groupG, letG* = Hom(G, A) and for anyE-moduleM, letM* = HomE(M, A). The evaluation map σG: G → G** is defined in the usual way andGisA-reflexive if σGis an isomorphism. This is strongly related to the question of whetherAis slender as anE-module, and we discuss thep-groups for which this holds. In some important cases,G** can be viewed as the completion ofGin a linear topology. It is known that ifA = circled plus,n Zpn, andGis ap-group of non-measurable cardinality, thenG** can be identified with the completion ofGin the circled plusc-topology, and we provide a generalization of this result. We also show that for any groupNof non-measurable cardinality there is a groupGsuch thatG**/σG(G) congruent with N.