Let
R
R
be a semiprime
K
K
-algebra with unity,
d
d
a nonzero derivation of
R
R
, and
f
(
x
1
,
…
,
x
t
)
f({x_1}, \ldots ,{x_t})
a monic multilinear polynomial over
K
K
such that
d
(
f
(
a
1
,
…
,
a
t
)
)
≠
0
d(f({a_1}, \ldots ,{a_t})) \ne 0
for some
a
1
,
…
,
a
t
∈
R
{a_1}, \ldots ,{a_t} \in R
. It is shown that if for every
r
1
,
…
,
r
t
{r_1}, \ldots ,{r_t}
in
R
R
either
d
(
f
(
r
1
,
…
,
r
t
)
)
=
0
d(f({r_1}, \ldots ,{r_t})) = 0
or
d
(
f
(
r
1
,
…
,
r
t
)
)
d(f({r_1}, \ldots ,{r_t}))
is invertible in
R
R
, then
R
R
is either a division ring
D
D
or
M
2
(
D
)
{M_2}(D)
, the ring of
2
×
2
2 \times 2
matrices over
D
D
, unless
f
(
x
1
,
…
,
x
t
)
f({x_1}, \ldots ,{x_t})
is a central polynomial for
R
R
. Moreover, if
R
=
M
2
(
D
)
R = {M_2}(D)
, where
2
R
≠
0
2R \ne 0
and
f
(
x
1
,
…
,
x
t
)
f({x_1}, \ldots ,{x_t})
is not a central polynomial for
D
D
, then
d
d
is an inner derivation of
R
R
.