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. Ren (2026) supplies an accompanying Lean development stating the counterexample
with the module-theoretic definitions. 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 ![[x_(i+1),x_i]=1](/images/equations/PureSemisimplicityConjecture/Inline10.svg)
and ![[x_i,x_j]=0](/images/equations/PureSemisimplicityConjecture/Inline11.svg)
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.
