What was already known
The conjecture is known for Artin algebras and several other classes. Simson’s division-ring criterion and later work of García and Šaroch reduced a counterexample program to controlling an infinite sequence of dual bimodules. Earlier finite-period constructions did not supply the required infinite sequence.
AI involvement
Ren explicitly credits GPT-6 Astra with developing the main proof and, in particular, the division rings T and H_r. Ren independently checked all proofs and completed the formalization and manuscript with GPT-6.1 Sol and Claude Opus 5.5. This documents a central model contribution within a human-led project.
Importance
Provides an explicit counterexample to a longstanding module-decomposition conjecture and separates right from left pure semisimplicity. The cited literature establishes the earlier criterion and unsuccessful construction program; no wider field-wide impact is assumed.
Limitations
The result is a preprint. Ren reports the human checks and Lean verification; no independent expert review is named. The formalization quantifies over modules in an arbitrary fixed universe and represents right modules through the opposite ring. Its proof that R is not left pure semisimple uses an explicit module and a countability argument instead of formalizing Chase’s general theorem. These proof choices and correspondence remain stated in the artifact.
Technical verification
Ren explicitly reports independently verifying every proof. The pinned artifact records a successful build and axiom audit of 832 theorems covering all sixteen labelled manuscript statements, including the actual triangular ring and its module decompositions. The main theorem uses only propext, Classical.choice and Quot.sound. The manuscript also reports a fresh Lean-kernel replay of all imported declarations. These are author-reported checks, with no independent expert review named.
Formal checking
Ren explicitly reports independently verifying every proof. The pinned artifact records a successful build and axiom audit of 832 theorems covering all sixteen labelled manuscript statements, including the actual triangular ring and its module decompositions. The main theorem uses only propext, Classical.choice and Quot.sound. The manuscript also reports a fresh Lean-kernel replay of all imported declarations. These are author-reported checks, with no independent expert review named.
- Checker
- Junpeng Ren and the author’s Lean verification tooling
- Scope
- 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.
- Basis
- Reported by the linked checker
Formal checking
Ren explicitly reports independently verifying all proofs; this is an author check, not independent external review.
- Checker
- Junpeng Ren (author)
- Scope
- 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.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Ren explains the mechanism in Sections 4–6: right bases and adjacent two-sided flags control the iterated duals, while coprime right indices force infinite left dimension. Quantum Laurent relations realize the required flags through explicit invertible coordinate matrices. His separate checking declaration supports his account without establishing independent community review.
Correction and publication history
Newly reviewed contribution with exact primary attribution and checking evidence.
Read this published revision