A construction of right Artinian right serial rings. (Q2847704)

The aim of the paper is the reduction of right Artinian, right serial rings to -- and their construction from -- a system \(X\), consisting of local right Artinian, right serial rings \(R_1,\ldots,R_n\) and \((R_i,R_j)\)-bimodules \(M_{ij}\) which correspond to the vertices, resp. edges, of a connected quiver \(Q\), together with a number of relations.NEWLINENEWLINE Basic to the construction is the tensor ring \(T(L)=\bigoplus_{\kappa\geq 0}L^{\otimes\kappa}\) of the \((R',R')\)-bimodule \(L=\bigoplus_{i,j}M_{ij}\) with \(R'=\bigoplus_iR_i\).NEWLINENEWLINE It is proved that for a particular ideal \(H\) of \(T(L)\), the quotient ring \(R(X)=T(L)/H\) is (*) indecomposable, right Artinian, right serial. Moreover, \(R(X)\) is universal in the following sense: If a (non-local) ring \(R\) is of type (*) and \(X'=X(R)\) is the system associated with \(R\), then the basic ring \(R^\circ\) of \(R\) is a homomorphic image of \(R(X')\).NEWLINENEWLINE In case of Artinian serial rings (generalized uniserial rings) the system \(X\) is essentially a VPE-system \(S\), introduced already by the reviewer [Arch. Math. 17, 20-35 (1966; Zbl 0135.07503)] for the construction and classification of all Artinian serial rings in terms of local ones.