Characterization of the centrally finite Amitsur-Small division rings
Abstract
In 1978, Amitsur and Small asked whether, for every division ring D, maximal left ideals of D[x1,…,xn] contract to maximal left ideals in smaller polynomial subrings. Chapman and the author showed that the answer is negative in general and called D an Amitsur-Small ring when this contraction property always holds. Earlier work showed that Hamilton's real quaternion algebra is Amitsur-Small, that division algebras of degree three are not Amitsur-Small, and that degree-two examples are restricted to a specific quaternionic form; subsequent work excluded cyclic division algebras of odd prime degree. We give a complete resolution in the centrally finite case. If D has finite dimension over its center F, then D is an Amitsur-Small ring if and only if either D=F, or F is real closed and D is the Hamilton quaternion algebra (−1,−1)F. In every other noncommutative centrally finite case, failure already occurs in two variables: there is a maximal left ideal M⊆ D[x,y] such that M∩ D[x] is not maximal in D[x].
Related
BibTeX
@misc{paran2026characterization,
title = {{Characterization of the centrally finite Amitsur-Small division rings}},
author = {Elad Paran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04317},
eprint = {2610.04317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04317},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characterization of the centrally finite Amitsur-Small division rings
AU - Elad Paran
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04317
UR - https://arxiv.org/abs/2610.04317
AB - In 1978, Amitsur and Small asked whether, for every division ring D, maximal left ideals of D[x_1,…,x_n] contract to maximal left ideals in smaller polynomial subrings. Chapman and the author showed that the answer is negative in general and called D an Amitsur-Small ring when this contraction property always holds. Earlier work showed that Hamilton's real quaternion algebra is Amitsur-Small, that division algebras of degree three are not Amitsur-Small, and that degree-two examples are restricted to a specific quaternionic form; subsequent work excluded cyclic division algebras of odd prime degree. We give a complete resolution in the centrally finite case. If D has finite dimension over its center F, then D is an Amitsur-Small ring if and only if either D=F, or F is real closed and D is the Hamilton quaternion algebra (−1,−1)_F. In every other noncommutative centrally finite case, failure already occurs in two variables: there is a maximal left ideal M⊆ D[x,y] such that M∩ D[x] is not maximal in D[x].
ER -