The Baer-Kaplansky theorem for almost completely decomposable groups (Q2738746)

The general question is to what degree the endomorphism ring of an algebraic object determines the object. There are many non-isomorphic subgroups of the additive group of rational numbers \(\mathbb{Q}\) that have isomorphic endomorphism rings. However, the groups, called of ring type, that are not only subgroups but subrings of \(\mathbb{Q}\) are isomorphic to their endomorphism rings and therefore the endomorphism ring determines the isomorphism class of such a subgroup. For torsion-free Abelian groups in general the problem is hopeless.NEWLINENEWLINENEWLINEThe authors first prove Theorem~2.1. Let \(X\), \(Y\) be completely decomposable groups with summands of ring type and \(\Theta\colon\text{End }X\to\text{End }Y\) a ring isomorphism. Then there is a group isomorphism \(\theta\colon X\to Y\) such that for all \(f\in\text{End }X\), \(f\Theta=\theta^{-1}f\theta\).NEWLINENEWLINENEWLINEThis is a ``strong Baer-Kaplansky type theorem'' which is possible due to the well-known and easy classification of completely decomposable groups up to isomorphism. Classification of torsion-free Abelian groups up to isomorphism is rarely possible, but there is an important weakening of isomorphism, near-isomorphism, and certain classes of almost completely decomposable groups can be classified up to near-isomorphism. The class of block-rigid crq-groups and its subclass of rigid crq-groups are examples which the authors exploit.NEWLINENEWLINENEWLINETheorem~3.6. Let \(X\) be a rigid crq-group of ring type and \(Y\) any crq-group. Then \(\text{End }X\cong\text{End }Y\) if and only if \(X\) is nearly isomorphic with \(Y\).NEWLINENEWLINENEWLINEThe proof uses that a rigid crq-group \(X\) is nearly isomorphic to the additive group of its endomorphism ring. The authors generalize one direction of 3.6 by a general trick.NEWLINENEWLINENEWLINETheorem~4.6. Let \(X\), \(Y\) be block-rigid crq-groups of ring type. Then \(X\) is nearly isomorphic with \(Y\) if \(\text{End }X\cong\text{End }Y\).NEWLINENEWLINENEWLINE\textit{W.~Lewis} [PhD. Dissertation, Honolulu (1992)] produced examples of almost completely decomposable groups that are nearly isomorphic but whose endomorphism rings are not isomorphic. These examples are almost completely decomposable groups with two critical types and are not crq-groups. The authors conjecture that the converse of 4.6 is true but do not resolve the question.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].