Let
R
R
be a commutative ring,
V
V
a finitely generated free
R
R
-module and
G
≤
G
L
R
(
V
)
G\le GL_R(V)
a finite group acting naturally on the graded symmetric algebra
A
=
S
(
V
)
A=S(V)
. Let
β
(
V
,
G
)
\beta (V,G)
denote the minimal number
m
m
, such that the ring
A
G
A^G
of invariants can be generated by finitely many elements of degree at most
m
m
.
For
G
=
Σ
n
G=\Sigma _n
and
V
(
n
,
k
)
V(n,k)
, the
k
k
-fold direct sum of the natural permutation module, one knows that
β
(
V
(
n
,
k
)
,
Σ
n
)
≤
n
\beta (V(n,k),\Sigma _n) \le n
, provided that
n
!
n!
is invertible in
R
R
. This was used by E. Noether to prove
β
(
V
,
G
)
≤
|
G
|
\beta (V,G) \le |G|
if
|
G
|
!
∈
R
∗
|G|! \in R^*
.
In this paper we prove
β
(
V
(
n
,
k
)
,
Σ
n
)
≤
m
a
x
{
n
,
k
(
n
−
1
)
}
\beta (V(n,k),\Sigma _n) \le max\{n,k(n-1)\}
for arbitrary commutative rings
R
R
and show equality for
n
=
p
s
n=p^s
a prime power and
R
=
Z
R = \mathbb {Z}
or any ring with
n
⋅
1
R
=
0
n\cdot 1_R=0
. Our results imply
β
(
V
,
G
)
≤
m
a
x
{
|
G
|
,
rank
(
V
)
(
|
G
|
−
1
)
}
\begin{equation*} \beta (V,G)\le max\{|G|, \operatorname {rank}(V)(|G|-1)\}\end{equation*}
for any ring with
|
G
|
∈
R
∗
|G| \in R^*
.