We extend a result of Rangaswamy about regularity of endomorphism rings of Abelian groups to arbitrary topological Abelian groups. Regularity of discrete quasi-injective modules over compact rings modulo radical is proved. A characterization of torsion LCA groups $A$ for which ${\mathrm{End}}_c\left(A\right)$ is regular is given.

Consider the four pairs of groups $(\Gamma ,R)$, $(\Gamma /{\Gamma }_0,R/{\Gamma }_0)$, $(K\cap S,P)$ and $(S,B)$, where $\Gamma$, $R$ are locally compact second countable abelian groups, $\Gamma$ is a dense subgroup of $R$ with inclusion map from $\Gamma$ to $R$ continuous; ${\Gamma }_0\subseteq \Gamma \subseteq R$ is a closed subgroup of $R$; $S$, $B$ are the duals of $R$ and $\Gamma$ respectively, and $K$ is the annihilator of ${\Gamma }_0$ in $B$. Let the first co-ordinate of each pair act on the second by translation. We connect, by a commutative diagram, the systems of imprimitivity which arise in a natural fashion on each pair, starting with a system...