We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of
n
×
n
n\times n
matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras
A
A
over infinite fields
k
k
of arbitrary characteristic. Our main result is the verification, for
A
A
, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal
P
P
of
A
A
is primitive if and only if the center of the Goldie quotient ring of
A
/
P
A/P
is algebraic over
k
k
, if and only if
P
P
is a locally closed point – with respect to the Jacobson topology – in the prime spectrum of
A
A
. These equivalences are established with the aid of a suitable group
H
\mathcal {H}
acting as automorphisms of
A
A
. The prime spectrum of
A
A
is then partitioned into finitely many “
H
\mathcal {H}
-strata” (two prime ideals lie in the same
H
\mathcal {H}
-stratum if the intersections of their
H
\mathcal {H}
-orbits coincide), and we show that a prime ideal
P
P
of
A
A
is primitive exactly when
P
P
is maximal within its
H
\mathcal {H}
-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which
H
\mathcal {H}
acts transitively on the set of rational ideals within each
H
\mathcal {H}
-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of
A
A
. For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.