Chain conditions in endomorphism rings (Q1073152)

A module over a ring R has finite left Goldie dimension if every direct sum of non-zero submodules has only finitely many summands. Let A be an abelian group and E(A) be the ring of its endomorphisms. An abelian group G is called a Goldie-group if E(G) is a left Goldie-ring, i.e. it has the ascending chain condition for left annihilators and finite left Goldie- dimension. An abelian group G is called quasi-Goldie, if E(G) satisfies ACC for left annihilators and for any subgroup \(U\subseteq G\) the condition \(G/U=\oplus_{i\in I}U_ i\) implies \(Hom_ Z(U_ i,G)=0\) for all but finitely many \(i\in I.\) The author considers the structure of Goldie- and quasi-Goldie-groups. He proves for an abelian group A the equivalence of the following conditions: (a) A is a (quasi) Goldie-group; (b) \(A=B\oplus T\oplus D\oplus E\) where B is a torsion-free reduced (quasi-) Goldie-group, T is finite, D is a divisible torsion group of finite rank such that: (i) \(T_ p\neq 0\) implies B/pB is finite, (ii) \(E_ p\neq 0\) implies \(B=pB\) and (iii) \(D\oplus E\neq 0\) implies that B has finite rank. Let A be a Goldie-group with E(A) semiprime. If A is non-singular as an E(A)-module then the functor \(Hom_ Z(A\),-) preserves direct sums of copies of A. An example of such group A having not finite rank over its center is obtained.