On mixed LCA groups with commutative rings of continuous endomorphisms

Let $\mathcal L$ be the class of locally compact abelian (LCA) groups. For $X\in\mathcal L$, let $E(X)$ denote the ring of continuous endomorphisms of $X$. In this paper, we determine for certain subclasses$\mathcal S$ of $\mathcal L$ the groups $X\in\mathcal S$ such that $E(X)$ is commutative. The main results concern the case of mixed LCA groups.