Separable torsion-free Abelian \(E^*\)-groups
The first half of this paper characterizes the torsion-free separable abelian groups \(G\) whose endomorphism semigroup \(E(G)^*\) admits a unique addition; that is, the endomorphism ring \(E(G)\) is isomorphic to any ring \(S\) for which \(E(G)^*\) is isomorphic to \(S^*\). It is shown that a necessary and sufficient condition is that the group \(G\) is semiconnected, that is, \(G\) has rank \(>1\) and a complementary summand of any rank 1 summand \(A\) has a rank 1 summand comparable with \(A\). The second half concerns \(E^*\)-groups, that is, torsion-free separable groups \(G\) such that for all rings \(R\) with \(R^*\) isomorphic to \(E(G)^*\), there is an abelian group \(H\) with \(E(H)\) isomorphic to \(R\). It is shown that a rank 1 group is an \(E^*\)-group if and only if it is divisible by only finitely many primes; and in general, \(G\) is an \(E^*\)-group if and only if every rank 1 summand which is not semiconnected is divisible by only finitely many primes.
scientific article; zbMATH DE number 1308261