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An Artin problem for division ring extensions and the pure semisimplicity conjecture. I. (English) Zbl 0873.16010

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 \(\operatorname{Hom}_G(_F(-)_G,G)\) 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.

MSC:

16G10 Representations of associative Artinian rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16K40 Infinite-dimensional and general division rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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