Prescribing endomorphism algebras of \(\aleph_n\)-free modules. (Q479507)

This 40-page technically demanding paper concerns the realisation of a given \(R\)-algebra \(A\) over a countable principal ideal domain \(R\) as the endomorphism ring \(E_R(G)\) of an \(\aleph_k\)-free \(R\)-module \(G\), subject to various necessary restrictions on the cardinalities of both \(A\) and \(G\). The difference between this result and others in the area lies in the relaxation of the index of \(\aleph_k\) from 1 to an arbitrary finite ordinal \(k\). The main tool, and the reason for the length of the paper, is an elaborate combinatorial construction known as a Black Box which is required to ensure that \(G\) is \(\aleph_k\)-free, but admits only endomorphisms from \(A\). By a suitable choice of \(A\), the authors are able to construct such \(G\) satisfying any of the properties: \(G\) is indecomposable; \(G\) is superdecomposable, i.e., \(G\) has no indecomposable summand; \(G\) satisfies Kaplansky's Test Problem, i.e., for any positive integer \(q\), \(G^s\cong G^r\) if and only if \(s\equiv r\pmod q\); \(G\) has a finite automorphism group.