Matrix representations of endomorphism rings for torsion-free abelian groups (Q6060228)

All groups studied in this paper are torsion-free abelian. The authors consider the so-called block-rigid crq-groups. They extend the combinatorial theory of ring type block-rigid crq-group direct decompositions to their endomorphism rings. The main result is the next theorem: Theorem. Let \(X\) be a block-rigid crq-group of ring type with critical typeset \(T = T_{cr}(X)= \{\tau_i : i = 1, \dots, k\}\), regulator \(A =\bigoplus_{i=1,2,\dots,k}A_{\tau_i}\), the regulator index \(e\) and \(n_i = \mathrm{rk}\,A_{\tau_i}\). If \(X\) admits a direct decomposition \(X = X_1 \oplus X_2 \oplus\dots\oplus X_s\) with indecomposable \(X_f\), then there exists a decomposition \(E = L_1 \oplus\dots\oplus L_s\) of the matrix ring \(E \cong \mathrm{End}\,X\) into the sum of indecomposable left ideals such that \(L_f^{+} \cong_{nr} X_f \oplus(\bigoplus_{i:\tau_i\in T_{cr}(X_f)}A_i^{'})\) where \(A_i^{'}\cong \tau_i^{n_i-1}\), \(f = 1, \dots ,s\). This leads to the combinatorial constructions of isomorphisms between non-commutative differently decomposable ring structures.