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PreprintCC BY 4.0arXiv

Mal'cev products of rings satisfying xn≈ x and idempotent semirings

  • Aifa Wang
  • Lili Wang
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Abstract

Let Rn be the variety of rings satisfying xneqid x, where n≥2, and let W be a variety of idempotent semirings with commutative addition. The equality Rn∘W=Rn∨W holds, and this variety has subvariety lattice L(Rn)× L(W). Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary W≤Slp.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026malcev,
  title = {{Mal'cev products of rings satisfying $x^n\approx x$ and idempotent semirings}},
  author = {Aifa Wang and Lili Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12188},
  eprint = {2610.12188},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12188},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Mal'cev products of rings satisfying x^n≈ x and idempotent semirings
AU  - Aifa Wang
AU  - Lili Wang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12188
UR  - https://arxiv.org/abs/2610.12188
AB  - Let Rn be the variety of rings satisfying x^neqid x, where n≥2, and let W be a variety of idempotent semirings with commutative addition. The equality Rn∘W=Rn∨W holds, and this variety has subvariety lattice L(Rn)× L(W). Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary W≤Slp.
ER  -