Let
R
R
be a prime algebra over a commutative ring
K
K
with unity and let
f
(
X
1
,
…
,
X
n
)
f(X_{1}, \ldots , X_{n})
be a multilinear polynomial over
K
K
. Suppose that
d
d
is a nonzero derivation on
R
R
such that for all
r
1
,
…
,
r
n
r_{1}, \ldots , r_{n}
in some nonzero ideal
I
I
of
R
R
,
[
d
(
f
(
r
1
,
…
,
r
n
)
)
,
f
(
r
1
,
…
,
r
n
)
]
k
=
0
\Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0
with
k
k
fixed. Then
f
(
X
1
,
…
,
X
n
)
f(X_{1}, \ldots , X_{n})
is central–valued on
R
R
except when char
R
=
2
R=2
and
R
R
satisfies the standard identity
s
4
s_{4}
in 4 variables.