Generalizations of isomorphism in torsion-free Abelian groups (Q2738742)

Abelian group theorists have long used equivalence relations weaker than isomorphism in studying torsionfree Abelian groups of finite rank (tffr groups), particularly stable isomorphism, near-isomorphism, and quasi-isomorphism. Recently, the authors placed the notion of multiple isomorphism into this toolchest [J. Algebra 221, No. 2, 536-550 (1999; Zbl 0945.20032)] -- tffr groups \(A\) and \(B\) are said to be multiple isomorphic provided \(A^n\cong B^n\) for all \(n\geq 2\). In the present paper, they survey results, applications, and connections involving these concepts. In particular, there is a hierarchy (isomorphism) \(\implies\) (multiple isomorphism) \(\implies\) (stable isomorphism) \(\implies\) (near-isomorphism) \(\implies\) (quasi-isomorphism), and none of the implications is reversible. The authors have also introduced [ibid.] a \(K_0\)-like invariant \(G(A)\) for any tffr group \(A\), which arises as follows: the Archimedean component of the multiple isomorphism class of \(A\) within the monoid of multiple isomorphism classes of tffr groups is a cancellative Abelian semigroup, and \(G(A)\) is its universal group. Various invariants of this group measure the ``relative coarseness'' of the above equivalence relations -- for example, the cardinality of the torsion subgroup of \(G(A)\) gives the number of multiple isomorphism classes of tffr groups near-isomorphic to \(A\). The torsionfree rank of \(G(A)\) gives the number of generators of \(G(A)\) up to near-isomorphism, and the authors obtain the following result in the rank 1 case: If \(A\) and \(B\) are strongly indecomposable tffr groups, each a direct summand of a direct sum of copies of the other, and if \(G(A)\) has torsionfree rank 1, then \(A\) and \(B\) are near-isomorphic.NEWLINENEWLINENEWLINEThe paper also includes a discussion of parallels between these ideas and recent work on von Neumann regular rings and exchange rings, and a selection of open problems. In the meantime, the authors have solved one of the fundamental problems concerning the new invariant \(G(A)\) -- they have shown that any finite Abelian group can appear as the torsion subgroup of a \(G(A)\), and that the torsionfree rank of \(G(A)\) can be arbitrarily large [\(K_0\)-like constructions for almost completely decomposable groups, preprint (2001)].NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].