On LCA groups whose rings of continuous endomorphisms have at most two non-trivial closed ideals. I. (Q2895944)

Let \(A\) be a nonzero locally compact abelian topological group (briefly, LCA-group) and \(\mathrm{End}_c(A)\) be the ring of continuous endomorphisms endowed with the compact-open topology. The author attempts to classify LCA-groups \(A\) for which the number of nontrivial closed ideals of \(\mathrm{End}_c(A)\) is \(0,1\) or \(2\), respectively.NEWLINENEWLINESome properties of LCA-groups for which the number of nontrivial closed ideals of \(\mathrm{End}_C(A)\) is \(0,1\) or \(2\), respectively, are derived. A complete classification of torsion LCA-groups whose rings of continuous endomorphisms have no nontrivial closed ideals is given. Furthermore, the author derives some properties of LCA-groups of exponent \(p^2\) or \(p^3, p\) prime, whose rings of continuous endomorphisms have two nontrivial closed ideals.NEWLINENEWLINERing properties of the ring \(\mathrm{End}(A)\) of endomorphisms of an abelian group without topology are studied in the books [\textit{L. Fuchs}, Infinite abelian groups. Vol. II. Pure and Applied Mathematics, 36. New York-London: Academic Press. (1973; Zbl 0257.20035)]; [\textit{P. A. Krylov, A. V. Mikhalev} and \textit{A. A. Tuganbaev}, Endomorphism rings of abelian Groups. Algebras and Applications 2. Boston MA: Kluwer Academic Publishers. (2003; Zbl 1044.20037)].