Let
R
R
be a nonassociative ring of characteristic different from
2
2
and
3
3
which satisfies the following identities:
\[
(
i)
(
a
b
,
c
,
d
)
+
(
a
,
b
,
[
c
,
d
]
)
=
a
(
b
,
c
,
d
)
+
(
a
,
c
,
d
)
b
,
({\text {i)}}\;(ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b,
\]
\[
(
ii)
(
a
,
a
,
a
)
=
0
,
({\text {ii)}}\;(a,a,a) = 0,
\]
\[
(
iii)
(
a
,
b
∘
c
,
d
)
=
b
∘
(
a
,
c
,
d
)
+
c
∘
(
a
,
b
,
d
)
({\text {iii)}}\;(a,b \circ c,d) = b \circ (a,c,d) + c \circ (a,b,d)
\]
for all
a
,
b
,
c
,
d
∈
R
a,b,c,d \in R
and with
x
∘
y
=
(
x
y
+
y
x
)
/
2
x \circ y = (xy + yx)/2
. We prove that if
R
R
is semiprime, then
R
R
is alternative.