Separable torsion-free abelian E*-groups Original Research Article

A ring R is said to be a unique addition ring (UA-ring) if any semigroup isomorphism R* = (R, *) not, vert, similar- (S, *) = S* of multiplicative semigroups with another ring S is always a ring isomorphism. See [5, 7–9] for earlier work on UA-rings. Depending on the context, we may or may not regard 0 as an element of R*. An abelian group G is called a UA-group if its endomorphism ring E(G) is a UA-ring. Given an abelian group G, denote by E*(G) the semigroup of all endomorphisms of G and let RG be the collection of all rings R such that R* not, vert, similar- E*(G). The group G is said to be an E*-group if for every ring (E*(G), circled plus), where circled plus is an addition on the semigroup E*(G), there is an abelian group H such that (E*(G), circled plus) is (isomorphic to) the endomorphism ring of H. Equivalently, G is an E*-group if for every ring R in RG there is an abelian group H such that R is (isomorphic to) the endomorphism ring of H.