Let L be the class of locally compact abelian groups. For X 2 L; let E(X) denote the ring of continuous endomorpgisms of X and t(X) the torsion subgroup of X: Given a prime p and a positive integer n; we denote by Z(pn) the cyclic group of order pn; by Z(p1) the quasi-cyclic group corresponding to p; and by Jp the group of p-adic integers, all taken discrete. Further, we denote by J¤ p the character group of Jp; by Zp the group of p-adic integers with its unique compact topology, and by Qp the group of p-adic numbers with its usual locally compact topology. Theorem 1. Let X be a residual group in L: The following conditions are equivalent: (i) E(X) is a field. (ii) E(X) is local and p1X is invertible in E(X) for all primes p: (iii) E(X) is local and X is densely divisible and torsionfree. (iv) X is topologically isomorphic with Qp for some prime p: A group X 2 L is said to be purely topologically indecomposable if every closed pure subgroup of X is topologically indecomposable. Theorem 2. Let X be a residual, purely topologically indecomposable group in L such that t(X) is closed in X: The following conditions are equivalent: (i) E(X) is local and commutative. (ii) E(X) is local. (iii) X is topologically isomorphic either with a pure subgroup G of Jp satisfying soc(G) = G; or with one of the groups Qp; Z(p1); or Z(pn); where p 2 P and n 2 N:A group X 2 L is said to be co-purely topologically indecomposable if for any closed pure subgroup C of X; the quotient group X=C is topologically indecomposable. Theorem 3. Let X be a residual, co-purely topologically indecomposable group in L such that mX=\n2N+nX is compact for some m 2 N+: The following conditions are equivalent: (i) E(X) is local and commutative. (ii) E(X) is local. (iii) X is topologically isomorphic either with a quotient group of J¤ p by a closed pure subgroup A satisfying rad(A) = f0g; or with one of the groups Qp; Zp; or Z(pn); where p 2 P and n 2 N: