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Left Jacobson Rings

J. Cimprič, M. Schötz

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 F-algebra A the ring A[$x_1$,...,$x_n$] of polynomials with coefficients in A is strongly left Jacobson and every maximal left ideal of A[$x_1$,...,$x_n$] 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.

Left Jacobson Rings

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 F-algebra A the ring A[,...,] of polynomials with coefficients in A is strongly left Jacobson and every maximal left ideal of A[,...,] 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.

Paper Structure

This paper contains 10 sections, 27 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. A brief survey of left Nulstellensätze
  3. Quaternionic Nullstellensatz
  4. Matrix Nullstellensatz
  5. Bergman's Nullstellensatz
  6. Geometric interpretation
  7. Counterexamples
  8. Some weakly Jacobson rings
  9. Some strongly Jacobson rings
  10. Polynomials with coefficients in algebras

Key Result

Theorem 1

Let $P$ be a subset of $\mathbb{H}[x_1,\ldots,x_d]$. For every element $q \in \mathbb{H}[x_1,\ldots,x_d]$ the following are equivalent.

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • Proposition 5
  • Proposition 6
  • Theorem 7
  • proof
  • Lemma 8
  • ...and 43 more