An Artin problem for division ring extensions and the pure semisimplicity conjecture. I
A ring \(R\) is right pure semisimple if every right \(R\)-module is a direct sum of finitely generated modules and it is of finite representation type if, further, there are just finitely many indecomposable modules up to isomorphism. The pure semisimplicity conjecture is that every right pure semisimple ring is of finite representation type. The author shows that the following would provide a counterexample to the pure semisimplicity conjecture: division rings \(F\subseteq G\) such that \(\dim(_FG)\) is infinite, \(\dim(G_F)=2\) and such that the right dimension of each of the iterated right dualisations of \(_FG_G\) with respect to \(\Hom_G(_F(-)_G,G)\) is 2. If there exist such \(F\subseteq G\) then the hereditary ring \(R_G=\left(\begin{smallmatrix} F & _FG_G \\ 0 & G\end{smallmatrix}\right)\) would have the following properties, among others: \(R_G\) would be right pure semisimple; the intersection of the finite powers of the radical of \(\text{mod-}R_G\) would be non-zero (and hence \(R_G\) would not be of finite representation type) but its square would be zero; for each \(m\geq 2\), the number of isomorphism classes of indecomposable right \(R_G\)-modules of length \(m\) would be 0 or 1.
direct sums of finitely generated modules
scientific article; zbMATH DE number 937055
scientific article