Pure semisimplicity and finite representation type
If every right module over a ring is a direct sum of finitely generated modules, must the ring have finite representation type? Such a ring is called right pure semisimple. Finite representation type means that, on both sides, the ring is artinian and has only finitely many indecomposable finitely generated module types. The conjecture asks whether right pure semisimplicity forces this stronger condition, and hence left pure semisimplicity.
Why it matters and what counts as progress
Why it matters: The question connects decompositions of arbitrary modules with classification by finitely many building blocks. A counterexample would show that the right and left module theories of one ring can have fundamentally different decomposition properties.
What counts as progress: Construct a right pure semisimple ring with infinite representation type, or prove that no such ring exists under stated hypotheses. A proposed construction must establish the module-decomposition property and the failure of finite representation type for the same ring.
How the problem developed
- 1976 · Special-case theorem [1]
Maurice Auslander Established the pure semisimplicity conjecture for Artin algebras. The arbitrary-ring question remained broader.
- 30 Sept 2019 · Counterexample program [2]
Jan Šaroch, following Daniel Simson and José Luis García Related possible counterexamples to division-ring embeddings and dimension sequences. Constructed a finite-period sequence and identified an unresolved route to the required infinite pattern.
Junpeng Ren with GPT-6 Astra Reported an explicit quantum-Laurent division-ring construction giving a right pure semisimple ring that is neither left pure semisimple nor of finite representation type. Ren reports checking every proof; the accompanying Lean artifact records the formal targets and audits.
Selected milestones, not a complete history. Earlier module-theoretic criteria and their authors retain credit; author-reported checking is distinct from independent expert review.
AI contribution
A counterexample to the pure semisimplicity conjecture
Ren constructs a division-ring extension H⊆T from nearest-neighbour quantum Laurent division rings with parameter 1/2. The bimodule has right dimension sequence (1,2,2,2,…) while T has infinite left dimension over H. The upper-triangular ring R=(H T; 0 T) is right pure semisimple, but neither left pure semisimple nor of finite representation type, giving a counterexample to the pure semisimplicity conjecture.