On LCA groups in which some closed subgroups have commutative rings of continuous endomorphisms

We describe here the locally compact abelian (LCA) groups all of whose closed polythetic, respectively, copolythetic subgroups have commutative rings of continuous endomorphisms. We also determine the LCA groups all of whose polythetic, respectively, copolythetic quotients by closed subgroups have commutative rings of continuous endomorphisms.