On hereditary rings and the pure semisimplicity conjecture II: Sporadic potential counterexamples

José L. García¹

¹ Department of Mathematics University of Murcia 30100 Murcia, Spain

Colloquium Mathematicum, Tome 139 (2015) no. 1, pp. 55-93

Résumé

$\hskip -1pt$It was shown in [Colloq. Math. 135 (2014), 227–262] that the pure semisimplicity conjecture (briefly, pssC) can be split into two parts: first, a weak pssC that can be seen as a purely linear algebra condition, related to an embedding of division rings and properties of matrices over those rings; the second part is the assertion that the class of left pure semisimple sporadic rings (ibid.) is empty. In the present article, we characterize the class of left pure semisimple sporadic rings having finitely many Auslander–Reiten components; the characterization is given through properties of the defining bimodules and the sequences of dimensions associated to these bimodules.

Détail

DOI : 10.4064/cm139-1-4

Keywords: hskip ptit shown colloq math pure semisimplicity conjecture briefly pssc split parts first weak pssc seen purely linear algebra condition related embedding division rings properties matrices those rings second part assertion class pure semisimple sporadic rings ibid empty present article characterize class pure semisimple sporadic rings having finitely many auslander reiten components characterization given through properties defining bimodules sequences dimensions associated these bimodules