Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 8 min ago
PreprintarXiv licencearXiv

Every commutative perfect ring is a quotient of an almost perfect domain

  • Xiaolei Zhang
Open at sourcePDFDOICite

Abstract

In this paper, we show that every nonzero commutative perfect ring is a quotient of a one-dimensional semilocal almost perfect domain of characteristic zero. The domain can be chosen to have as many maximal ideals as the given ring has local factors. The construction uses coefficient discrete valuation rings, triangular presentations of purely inseparable residue extensions, and approximation for finitely many independent rank-one valuations. Independent error variables keep the construction inside a field. A normal-form argument shows that the divided elements used to get perfect principal quotients do not kill any further good monomial. In positive residue characteristic, the choice a=2c+1 with c≥ n protects the nonzero powers of p in a target ring of characteristic pn. We also give local and mixed-characteristic consequences, and examples with non-nilpotent radicals.

Related

  1. ALMOST EVERY PAIR OF INTERVAL EXCHANGE TRANSFORMATIONS DOES NOT GENERATE A FREE GROUP
    Sébastien Gouëzel
    HALPreprintCC BY 4.0Full textDynamical systemsHALarXiv
  2. Almost Every Graph Is Reconstructible from Its Token Graphs
    Yuanming Luo
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  3. On almost perfect additive complements
    Wouter van Doorn, Boning Liu, Quanyu Tang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Almost perfect graph classes
    Cicely Henderson, Hidde Koerts, Taite LaGrange +4
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Almost every Boolean function has central threshold degree
    Martin Anthony
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{zhang2026every,
  title = {{Every commutative perfect ring is a quotient of an almost perfect domain}},
  author = {Xiaolei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24940},
  eprint = {2610.24940},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24940},
}
RIS
TY  - UNPB
TI  - Every commutative perfect ring is a quotient of an almost perfect domain
AU  - Xiaolei Zhang
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.24940
UR  - https://arxiv.org/abs/2610.24940
AB  - In this paper, we show that every nonzero commutative perfect ring is a quotient of a one-dimensional semilocal almost perfect domain of characteristic zero. The domain can be chosen to have as many maximal ideals as the given ring has local factors. The construction uses coefficient discrete valuation rings, triangular presentations of purely inseparable residue extensions, and approximation for finitely many independent rank-one valuations. Independent error variables keep the construction inside a field. A normal-form argument shows that the divided elements used to get perfect principal quotients do not kill any further good monomial. In positive residue characteristic, the choice a=2c+1 with c≥ n protects the nonzero powers of p in a target ring of characteristic p^n. We also give local and mixed-characteristic consequences, and examples with non-nilpotent radicals.
ER  -