Ore extensions of commutative rings and the Dixmier-Moeglin equivalence

Abstract

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