Representations of \(\mathcal D\)-forms
Let \(\mathcal D\) be a two sided Noetherian hereditary ring without zero- divisors, \(D\) be the division ring of fractions of \(\mathcal D\) and \(A = M_ n(D)\). A finite collection \(S = (A_ i,V_{ij})\), \(i,j = 1,\dots,n\), where \(A_ i =M_{n_ i} (D)\) and \(V_{ij}\) is a compatible \(A_ i\)- \(A_ j\)-bimodule, is called \(D\)-form. If for some \(i\) (distinguished vertices) \(A_ i\) are substituted in \(S\) by compatible \(D\)-orders \(\Lambda_ i\) of \(A_ i\), then the resulting collection is called \(\mathcal D\)-form. The scheme of a \(\mathcal D\)-form \(\Sigma=(B_ i,V_{ij})\) is the graph \(\Gamma(\Sigma)\) with the vertices \(1,2,\dots,n\) and \(d_{ij}\) arrows from \(i\) to \(j\), where \(d_{ij} = (\dim_ DV_{ij})/(n_ i,n_ j)\). A completion \(\widehat{\Gamma}(\Sigma)\) is obtained from \(\Gamma(\Sigma)\) by exchange of each distinguished vertex by an infinite chain. Let \(T(\Sigma)\) be the tensor algebra of the bimodule \(V = \bigoplus_{i,j} V_{ij}\) over the ring \(B = \prod_ iB_ i\). A \(\mathcal D\)-form \(\Sigma\) is of bounded module type if there is a natural number \(N\) such that every indecomposable finitely presented \(T(\Sigma)\)-module is generated by \(N\) elements. The main result is the following. A \(\mathcal D\)-form \(\Sigma\) is of bounded module type if and only if: 1) either \(\mathcal D\) is a simple ring and any connected component of the scheme \(\Gamma(\Sigma)\) is a Dynkin scheme; 2) or \(\mathcal D\) is not a simple ring and every finite connected component of the completed scheme \(\widehat{\Gamma}(\Sigma)\) is a Dynkin scheme.
indecomposable finitely presented \(T(\Sigma)\)- module
scientific article; zbMATH DE number 93098