TY - JOUR
T1 - Amitsur-Small rings
AU - Chapman, Adam
AU - Paran, Elad
N1 - Publisher Copyright:
© 2025 The Authors
PY - 2025/5/23
Y1 - 2025/5/23
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.
UR - http://www.scopus.com/inward/record.url?scp=105006784065&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2025.04.049
DO - 10.1016/j.jalgebra.2025.04.049
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AN - SCOPUS:105006784065
SN - 0021-8693
VL - 679
SP - 86
EP - 95
JO - Journal of Algebra
JF - Journal of Algebra
ER -