On topological endomorphism rings with no more than two non-trivial closed ideals

Abstract

Let L be the class of locally compact abelian groups. For X ∈ L, we denote by t(X) the torsion subgroup of X and by E(X) the ring of continuous endomorphisms of X, taken with the compact-open topology. If X is topologically torsion, then S(X) stands for the set of primes p such that the corresponding topological p-primary component of X is non-zero. Given a positive integer n, we set nX = {nx | x ∈ X} and X[n] = {x ∈ X | nx = 0}.