| Title: | Left Jacobson rings |
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| Authors: | ID Cimprič, Jaka (Author)
ID Schötz, Matthias (Author) |
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| Files: |  PDF - Presentation file, download (1,02 MB)
MD5: 2312A7BB8C9874A72E1C083CFD5E28A0
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0021869326001341
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | Nullstellensatz, noncommutative geometry, maximal left ideals, Jacobson ring, Azumaya algebra, Weyl algebra |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.07.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 453-472 |
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| Numbering: | Vol. 698 |
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| PID: | 20.500.12556/DiRROS-28954  |
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| UDC: | 512 |
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| ISSN on article: | 0021-8693 |
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| DOI: | 10.1016/j.jalgebra.2026.03.002 |
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| COBISS.SI-ID: | 275191043 |
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| Note: |
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| Publication date in DiRROS: | 14.04.2026 |
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| Views: | 26 |
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| Downloads: | 10 |
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| Metadata: |    |
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