In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every pure extension of G by a group in K splits and obtain a corresponding dual result.
Loth, Peter, "Pure Extensions of Locally Compact Abelian Groups" (2006). Mathematics Faculty Publications. Paper 24.