Left Jacobson rings

Cimprič, Jaka (avtor), Schötz, Matthias (avtor)

We say that a ring is strongly (resp. weakly) left Jacobson if every semiprime (resp. prime) left ideal is an intersection of maximal left ideals. There exist Jacobson rings that are not weakly left Jacobson, e.g. the Weyl algebra. Our main result is the following one-sided noncommutative Nullstellensatz: For any finite-dimensional ${\mathbb F}$-algebra ${\mathbb A}$ the ring ${\mathbb A}[x_1, \ldots,x_n]$ of polynomials with coefficients in ${\mathbb A}$ is strongly left Jacobson and every maximal left ideal of ${\mathbb A}[x_1, \ldots,x_n]$ has finite codimension. We also prove that an Azumaya algebra is strongly left Jacobson iff its center is Jacobson and that an algebra that is a finitely generated module over its center is weakly left Jacobson iff it is Jacobson.

• Jezik: Angleški jezik
• Ključne besede: Nullstellensatz, noncommutative geometry, maximal left ideals, Jacobson ring, Azumaya algebra, Weyl algebra
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.07.2026
• Leto izida: 2026
• Št. strani: Str. 453-472
• Številčenje: Vol. 698
• PID: 20.500.12556/RUL-181723
• UDK: 512
• ISSN pri članku: 0021-8693
• DOI: 10.1016/j.jalgebra.2026.03.002
• COBISS.SI-ID: 275191043
• Datum objave v RUL: 14.04.2026

Gradivo je del revije

• Naslov: Journal of algebra
• Skrajšan naslov: J. algebra
• Založnik: Elsevier
• ISSN: 0021-8693
• COBISS.SI-ID: 1310986

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.