Let
H
H
be a connected Hopf algebra acting on an algebra
A
A
. Working over a base field having characteristic
0
0
, we show that for a given prime (semi-prime, completely prime) ideal
I
I
of
A
A
, the largest
H
H
-stable ideal of A contained in
I
I
is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.