  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 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 stronger than the Dixmier-Moeglin
equivalence.
  For a simple complex Lie algebra g, a non root of unity q\neq 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 U_q[w] 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.
