Pure Semisimplicity Conjecture

The pure semisimplicity conjecture asserts that a right pure semisimple ring has only finitely many isomorphism classes of indecomposable right modules. All these modules are finitely generated in this setting. This is the finite representation type condition, and the conjecture is equivalent to requiring every right pure semisimple ring to be left pure semisimple (Šaroch 2021).

Ren (2026) reported a counterexample, an upper triangular ring

where is a division subring of a division ring and the upper-right entry is the natural - bimodule. Ren (2026) claims that is right pure semisimple but neither left pure semisimple nor of finite representation type. The expanded construction uses a parameter of infinite multiplicative order. Over , the choices , 3, 4, ... give pairwise Morita inequivalent rings. Parameters of finite order at least 2 instead yield rings of finite representation type. The accompanying Lean formalization covers only the original counterexample, not the finite-order family or the Morita inequivalence results. The claimed refutation remains subject to independent assessment. Ren (2026) reports GPT-6 Astra assistance with the construction and proof, and GPT-6.1 Sol and Claude Opus 5.5 assistance with formalization.

Generalov and Tursunbaev (2026) report another counterexample of this triangular form. They construct a division ring by adjoining a shift to the division ring of fractions of a rational algebra with generators for and relations and for . The shift satisfies . Taking the division subring generated over by and as the upper-left entry gives a hereditary ring claimed to be right pure semisimple with infinitely many isomorphism classes of indecomposable right modules.