On torsionfree LCA groups with commutative rings of continuous endomorphisms

Let $\mathcal L$ be the class of locally compact abelian (LCA) groups. For certain subclasses $\mathcal S$ of $\mathcal L$, we obtain information about the groups $X\in\mathcal S$ such that the ring $E(X)$ of continuous endomorphisms of $X$ is commutative. The main results concern torsionfree groups, groups with splitting torsion subgroups and their duals.