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GeezerVison is showing mathworld.wolfram.com — Pure Semisimplicity Conjecture -- from Wolfram MathWorld 
Pure Semisimplicity ConjectureThe 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
Generalov and Tursunbaev (2026) report another counterexample of this triangular form. They construct a division ring See alsoPure Semisimple Ring, Semisimple RingExplore with Wolfram|Alpha
More things to try: ReferencesGeneralov, A. I. and Tursunbaev, A. "Division Ring Extensions and a Counterexample to the Pure Semisimplicity Conjecture." 6 Nov 2026. https://arxiv.org/abs/2611.09749v1.Ren, J. "A Counterexample to the Pure Semisimplicity Conjecture." 29 Oct 2026. https://arxiv.org/abs/2610.39883.Šaroch, J. "Pure Semisimplicity Conjecture and Artin Problem for Dimension Sequences." J. Pure Appl. Algebra 225, 106745, 2021. https://doi.org/10.1016/j.jpaa.2021.106745.Cite this as:Weisstein, Eric W. "Pure Semisimplicity Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PureSemisimplicityConjecture.html Subject classifications | |
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is a division subring of a 
and the upper-right entry is the natural 
-
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.
by adjoining a shift 
to the 
for 
and relations ![[x_(i+1),x_i]=1](/api/image?u=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FPureSemisimplicityConjecture%2FInline10.svg){kind=small}
and ![[x_i,x_j]=0](/api/image?u=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FPureSemisimplicityConjecture%2FInline11.svg){kind=small}
for 
.
.
by 
and 
as the upper-left entry gives a hereditary