Dualities between Almost Completely Decomposable Groups and their Endomorphism Rings

Abstract

We prove that endomorphism rings of nearly isomorphic, almost completely decomposable groups of ring type are also nearly isomorphic as additive structures. On this basis, acd groups can be considered in a dual connection with their endomorphism rings.

Near Isomorphism for a Class of Infinite Rank Torsion-Free Abelian Groups

“…In this section we will prove that near isomorphism between bcd-groups of this structure induces near isomorphism of their endomorphism rings and reflects on the class by Theorem 3.5. This extends the results in Blagoveshchenskaya (2005) in some direction to infinite rank groups. We shall not consider near isomorphism for arbitrary bcd-groups and therefore leave the following question open.…”

“…Finally, we consider the endomorphism rings of groups from our class with regulating quotient a bounded p-group and prove that for such near isomorphic groups their endomorphism rings are also near isomorphic as additive groups, which is in analogy to the results proved by the first author in Blagoveshchenskaya (2005).…”

“…The proof needs more technical results which extend the approach in Blagoveshchenskaya (2005), in particular, Blagoveshchenskaya (2005, Lemma 3.1).…”

“…In order to extend the results from Blagoveshchenskaya (2005) we have to introduce some notation and to prove well-known results for almost completely decomposable groups (like type isomorphism) in a generalized form for bcd-groups. We shall pose some restrictions on our groups which are reasonable assumptions for our purposes.…”

Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

“…In contrast with this, some properties can be extended from groups X with primary X/R(X) to arbitrary acd groups. The fact that nearly isomorphic acd groups have nearly isomorphic endomorphism rings considered as torsion-free Abelian groups, was generalized from the case of primary regulator quotient groups (see [6]).…”