Amitsur-Small rings

Adam Chapman; Elad Paran

doi:10.1016/j.jalgebra.2025.04.049

Amitsur-Small rings

Adam Chapman, Elad Paran

Mathematics

Open University of Israel

Research output: Contribution to journal › Article › peer-review

Abstract

Let R_n=D[x₁,…,x_n] denote the ring of polynomials in n central variables over a division ring D. We say that D is an Amitsur-Small ring if for any maximal left ideal in R_n, M∩R_k is a maximal left ideal in R_k, for all n∈N and 1≤k≤n. We demonstrate the existence of non Amitsur-Small division rings, providing a negative answer to a question of Amitsur and Small from 1978. We show that Hamilton's real quaternion algebra H=(−1,−1)_2,R is an Amitsur-Small ring, division rings of degree 3 over their center F are never Amitsur-Small, and division rings of degree 2 are not Amitsur-Small if they are not quaternion algebras (−1,−1)_2,F over a Pythagorean field F.

Original language	English
Pages (from-to)	86-95
Number of pages	10
Journal	Journal of Algebra
Volume	679
DOIs	https://doi.org/10.1016/j.jalgebra.2025.04.049
State	Published - 23 May 2025

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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title = "Amitsur-Small rings",

abstract = "Let Rn=D[x1,…,xn] denote the ring of polynomials in n central variables over a division ring D. We say that D is an Amitsur-Small ring if for any maximal left ideal in Rn, M∩Rk is a maximal left ideal in Rk, for all n∈N and 1≤k≤n. We demonstrate the existence of non Amitsur-Small division rings, providing a negative answer to a question of Amitsur and Small from 1978. We show that Hamilton's real quaternion algebra H=(−1,−1)2,R is an Amitsur-Small ring, division rings of degree 3 over their center F are never Amitsur-Small, and division rings of degree 2 are not Amitsur-Small if they are not quaternion algebras (−1,−1)2,F over a Pythagorean field F.",

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AU - Paran, Elad

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N2 - Let Rn=D[x1,…,xn] denote the ring of polynomials in n central variables over a division ring D. We say that D is an Amitsur-Small ring if for any maximal left ideal in Rn, M∩Rk is a maximal left ideal in Rk, for all n∈N and 1≤k≤n. We demonstrate the existence of non Amitsur-Small division rings, providing a negative answer to a question of Amitsur and Small from 1978. We show that Hamilton's real quaternion algebra H=(−1,−1)2,R is an Amitsur-Small ring, division rings of degree 3 over their center F are never Amitsur-Small, and division rings of degree 2 are not Amitsur-Small if they are not quaternion algebras (−1,−1)2,F over a Pythagorean field F.

AB - Let Rn=D[x1,…,xn] denote the ring of polynomials in n central variables over a division ring D. We say that D is an Amitsur-Small ring if for any maximal left ideal in Rn, M∩Rk is a maximal left ideal in Rk, for all n∈N and 1≤k≤n. We demonstrate the existence of non Amitsur-Small division rings, providing a negative answer to a question of Amitsur and Small from 1978. We show that Hamilton's real quaternion algebra H=(−1,−1)2,R is an Amitsur-Small ring, division rings of degree 3 over their center F are never Amitsur-Small, and division rings of degree 2 are not Amitsur-Small if they are not quaternion algebras (−1,−1)2,F over a Pythagorean field F.

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M3 - مقالة

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VL - 679

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EP - 95

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Amitsur-Small rings

Abstract

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