Isomorphism problem for tensor algebras over valued graphs (Q1177980)

Let \(\Sigma\) be a valued graph with a modulation \((D,M)\). We denote by \(T(\Sigma)=T(\Sigma,D,M)\) the tensor ring over \(\Sigma\). The main result is the following theorem: Let \(\Sigma,\;\Sigma'\) be valued graphs with modulations \((D,M)\) and \((D',M')\), respectively. Then \(T(\Sigma)\) is isomorphic to \(T(\Sigma')\) if and only if \(\Sigma\) is isomorphic to \(\Sigma'\) and \((D,M)\) is isomorphic to \((D',M')\). As a consequence of this result, the author gives the following corollary which was proved by himself and others earlier: Two path algebras of quivers over fields are isomorphic if and only if the quivers are isomorphic and the ground fields are isomorphic. In the second part of the paper the author gives some necessary and sufficient conditions in terms of data of valued graphs for a tensor ring to be left artinian, left noetherian, primary, semiprime and prime, respectively. All proofs are elementary and elegant.