ADDITIVITY OF JORDANn-TUPLEDERIVABLE MAPS ON ALTERNATIVERINGS

Abstract

Let
R
be an alternative ring. We study the additivity of maps
δ
:
R
→
R
satisfying the following condition
δ
(
a
n
◦
(
···
(
a
2
◦
a
1
)
···
)) =
∑
n
k
=1
a
n
◦
(
···
(
δ
(
a
k
)
◦
(
···
(
a
2
◦
a
1
)
···
))
···
) for all
a
1
,
···
,a
n
∈
R
,
where
a
◦
b
=
ab
+
ba
is the Jordan product of
a
and
b
in
R
.
We prove that if
R
contains a non-trivial idempotent satisfying some conditions, then
δ
is
additive.