The paper concerns some cases of ring extensions
R
⊂
S
R \subset S
, where
S
S
is finitely generated as a right
R
R
-module and
R
R
is right Noetherian. In
\S
1
{\text {\S }}1
it is shown that if
R
R
is a Jacobson ring, then so is
S
S
, with the converse true in the
PI
{\text {PI}}
case. In
\S
2
{\text {\S }}2
we show that if
S
S
is semiprime
PI
{\text {PI}}
,
R
R
must also be left (as well as right) Noetherian and
S
S
is finitely generated as a left .
R
R
-module.
\S
3
{\text {\S }}3
contains a result on
E
E
-rings.