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Von Neumann regular and semisimple rings realized as principal quotients of domains

  • Xiaolei Zhang
  • Wei Qi
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Abstract

In this paper, we show that every commutative von Neumann regular ring is a quotient of a characteristic-zero Bézout domain by its principal Jacobson radical. We then characterize the commutative semisimple rings that occur as quotients of Dedekind domains. For a finite product of fields R=∏i=1n ki, such a presentation exists exactly when the cardinals |ki|ℵ0 are all equal. When this condition holds, the source may be taken to be a characteristic-zero semilocal principal ideal domain, and the quotient ideal may again be its Jacobson radical.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026neumann,
  title = {{Von Neumann regular and semisimple rings realized as principal quotients of domains}},
  author = {Xiaolei Zhang and Wei Qi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40185},
  eprint = {2610.40185},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40185},
}
RIS
TY  - UNPB
TI  - Von Neumann regular and semisimple rings realized as principal quotients of domains
AU  - Xiaolei Zhang
AU  - Wei Qi
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40185
UR  - https://arxiv.org/abs/2610.40185
AB  - In this paper, we show that every commutative von Neumann regular ring is a quotient of a characteristic-zero Bézout domain by its principal Jacobson radical. We then characterize the commutative semisimple rings that occur as quotients of Dedekind domains. For a finite product of fields R=∏_i=1^n k_i, such a presentation exists exactly when the cardinals |k_i|^ℵ_0 are all equal. When this condition holds, the source may be taken to be a characteristic-zero semilocal principal ideal domain, and the quotient ideal may again be its Jacobson radical.
ER  -