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arXiv · 2210.12024

Ore extensions of commutative rings and the Dixmier-Moeglin equivalence

Jason P. Bell·Léon Burkhardt·Nicholas Priebe
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Abstract

We consider Ore extensions of the form $T:=R[x;\sigma,\delta]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in {\rm Spec}(T)$ is primitive if and only if $\{P\}$ is locally closed in ${\rm Spec}(T)$, if and only if the Goldie ring of quotients of $T/P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$.

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Jason P. Bell, Léon Burkhardt, Nicholas Priebe. 2022-10-21. Ore extensions of commutative rings and the Dixmier-Moeglin equivalence. https://arxiv.org/abs/2210.12024

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