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OP-11358Status unassessedAlgebra

Generalising a result of Schmidt-van den Dries · Question 10.1

Noncommutative surfaces and Artin's conjecture · AIMPL workshop record

MSC 16 · Associative rings and algebras

Editorial mapping of AIMPL list topic: Noncommutative surfaces and Artin's conjecture. http://aimpl.org/ncsurfaceartin/ Not an author-supplied MSC code.

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Problem statement

Does this result have a noncommutative analogue? More precisely, does a similar result hold if we replace RR by k⟨x1,…,xd⟩/(r1,…,rs)k\langle x_1,\dots, x_d \rangle /(r_1, \dots, r_s)?

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Background

Question context

A result of Schmidt--van den Dries says the following:

Suppose that R=k[x1,…,xd]/(r1,…,rs)R = k[x_1,\dots, x_d]/(r_1, \dots, r_s) and there exists nn such that deg⁡(ri)≤n\deg(r_i) \leq n for all ii. Then there exists N=N(d,s,n)N = N(d,s,n) such that if RR is not a domain, then there exist nonzero p,q∈Rp,q \in R of degree less than NN such that pq=0pq = 0.

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References
  1. [1]Noncommutative surfaces and Artin's conjecture · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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