A counterexample to the pure semisimplicity conjecture

@misc{ren2026counterexample,
  title = {{A counterexample to the pure semisimplicity conjecture}},
  author = {Junpeng Ren},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39883},
  eprint = {2610.39883},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39883},
}

TY  - UNPB
TI  - A counterexample to the pure semisimplicity conjecture
AU  - Junpeng Ren
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39883
UR  - https://arxiv.org/abs/2610.39883
AB  - A ring is right pure semisimple if every right module is a direct sum of finitely generated modules, and the pure semisimplicity conjecture asserts that every such ring is of finite representation type. In this article, a division ring extension H⊆ T is constructed such that the bimodule _HT_T has right dimension sequence (1,2,2,2,…) and T is infinite-dimensional as a left H-space. The division ring T is the union of the fraction division rings of the nearest-neighbour quantum Laurent algebras with parameter 1/2, and H is the image of T under a compatible family of conjugations. The extension is controlled by right bases of every length and by two-sided flags of adjacent division subrings, which identify all iterated right duals up to explicit twists. It follows that the triangular ring (H&T 0&T) is right pure semisimple but neither left pure semisimple nor of finite representation type, so that the pure semisimplicity conjecture fails.
ER  -