Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier-Moeglin equivalence.We define quantities which measure how "close" an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally "close" to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier-Moeglin equivalence. Using the example of the universal enveloping algebra of sl 2 (C), we show that the strong Dixmier-Moeglin equivalence is strictly stronger than the Dixmier-Moeglin equivalence.For a simple complex Lie algebra g, a non root of unity q = 0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subalgebra U q [w] of the quantised enveloping K-algebra U q (g). These quantum Schubert cells are known to satisfy the Dixmier-Moeglin equivalence and we show that they in fact satisfy the strong Dixmier-Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier-Moeglin equivalence.• A prime ideal P of a noetherian K-algebra R is said to be rational if the field extension Z(Frac R/P ) of K is algebraic.Dixmier and Moeglin proved that for a prime ideal of the universal enveloping algebra of a finite-dimensional complex Lie algebra, the properties of being primitive, locally closed, and rational are equivalent. In modern terminology, they proved that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier-Moeglin equivalence.Since the work of Dixmier and Moeglin on universal enveloping algebras of finite-dimensional complex Lie algebras, many more algebras have been shown to satisfy the Dixmier-Moeglin equivalence: [4, Corollary II.8.5] lists several quantised coordinate rings which satisfy the Dixmier-Moeglin equivalence; the first named author, Rogalski, and Sierra [1] have shown that twisted homogeneous coordinate rings of projective surfaces satisfy the Dixmier-Moeglin equivalence. However, Irving [14] and Lorenz [17] have shown that there exist noetherian algebras for which the Dixmier-Moeglin equivalence fails.Our goal is to extend the notion of the Dixmier-Moeglin equivalence to all prime ideals, in a way which captures how "close" they are to being primitive. Of course, not all non-primitive prime ideals are created equal. For example, in the polynomial ring C[x, y], the primitive ideals are the maximal ideals x − α, y − β . For this reason, we think of the prime ideal x as being "closer" to being primitive than the prime ideal 0 , in the same sense that it...
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