  Let $k$ be a field and let $R$ be a left noetherian $k$-algebra. The algebra
$R$ satisfies the Dixmier-Moeglin equivalence if the annihilators of
irreducible representations are precisely those prime ideals that are locally
closed in the ${\rm Spec}(R)$ and if, moreover, these prime ideals are
precisely those whose extended centres are algebraic extensions of the base
field. We show that if $R$ and $S$ are two left noetherian $k$-algebras with
${\rm dim}_k(R), {\rm dim}_k(S)<|k|$ then if $R$ and $S$ have homeomorphic
spectra then $R$ satisfies the Dixmier-Moeglin equivalence if and only if $S$
does. In particular, the topology of ${\rm Spec}(R)$ can detect the
Dixmier-Moeglin equivalence in this case. In addition, we show that if $k$ is
uncountable and $R$ is affine noetherian and its prime spectrum is a disjoint
union of subspaces that are each homeomorphic to the spectrum of an affine
commutative ring then $R$ satisfies the Dixmier-Moeglin equivalence. We show
that neither of these results need hold if $k$ is countable and $R$ is
infinite-dimensional. Finally, we make the remark that satisfying the
Dixmier-Moeglin equivalence is a Morita invariant and finally we show that $R$
and $S$ are left noetherian $k$-algebras that satisfy the Dixmier-Moeglin
equivalence then $R\otimes_k S$ does too, provided it is left noetherian and
satisfies the Nullstellensatz; and we show that $eRe$ also satisfies the
Dixmier-Moeglin equivalence, where $e$ is a nonzero idempotent of $R$.
