Let
G
G
be a linearly reductive group over a field
k
k
, and let
R
R
be a
k
k
-algebra with a rational action of
G
G
. Given rational
R
R
-
G
G
-modules
M
M
and
N
N
, we define for the induced
G
G
-action on Hom
R
(
M
,
N
)
_{R}(M,N)
a generalized Reynolds operator, which exists even if the action on Hom
R
(
M
,
N
)
_{R}(M, N)
is not rational. Given an
R
R
-module homomorphism
M
→
N
M \rightarrow N
, it produces, in a natural way, an
R
R
-module homomorphism which is
G
G
-equivariant. We use this generalized Reynolds operator to study properties of rational
R
R
-
G
G
modules. In particular, we prove that if
M
M
is invariantly generated (i.e.
M
=
R
⋅
M
G
M = R \cdot M^{G}
), then
M
G
M^{G}
is a projective (resp. flat)
R
G
R^{G}
-module provided that
M
M
is a projective (resp. flat)
R
R
-module. We also give a criterion whether an
R
R
-projective (or
R
R
-flat) rational
R
R
-
G
G
-module is extended from an
R
G
R^{G}
-module.