Relationships between the prime ideals of a ring
R
R
and of a normalizing extension
S
S
have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of
R
R
and of
T
T
where
T
T
is a ring with
R
⊂
T
⊂
S
R \subset T \subset S
, and
S
S
is a normalizing extension of
R
R
: such rings
T
T
are called intermediate normalizing extensions of
R
R
. One result ("Cutting Down") asserts that for any prime ideal
J
J
of
T
T
,
J
∩
R
J \cap R
is the intersection of a finite set of prime ideals
P
i
{P_i}
of
R
R
, uniquely defined by
J
J
, whose corresponding factor rings
R
/
P
i
R/{P_i}
are mutually isomorphic. The minimal members of the family of
P
i
{P_i}
’s are the primes of
R
R
minimal over
J
∩
R
J \cap R
, and an "incomparability" theorem is proved which shows that no two comparable primes of
T
T
can have their intersections with
R
R
share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between
T
/
J
T/J
and each of the rings
R
/
P
i
R/{P_i}
, and a demonstration that the "additivity principle" holds.