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On hereditary artinian rings and the pure semisimplicity conjecture: rigid tilting modules and a weak conjecture

José L. García — 2014

Colloquium Mathematicae

A weak form of the pure semisimplicity conjecture is introduced and characterized through properties of matrices over division rings. The step from this weak conjecture to the full pure semisimplicity conjecture would be covered by proving that there do not exist counterexamples to the conjecture in a particular class of rings, which is also studied.

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

José L. García — 2015

Colloquium Mathematicae

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...

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