Affiliation:
1. Department of Mathematics, College of Science and Arts at Balgarn, University of Bisha , Sabt Al-Alaya(61985) , Saudi Arabia
2. Department of Mathematics, Faculty of Science, Al-Azhar University , Nasr City (11884) , Cairo , Egypt
Abstract
Abstract
In this article, we define the following: Let
N
0
{{\mathbb{N}}}_{0}
be the set of all nonnegative integers and
D
=
(
d
i
)
i
∈
N
0
D={\left({d}_{i})}_{i\in {{\mathbb{N}}}_{0}}
a family of additive mappings of a
∗
\ast
-ring
R
R
such that
d
0
=
i
d
R
{d}_{0}=i{d}_{R}
.
D
D
is called a Jordan
(
α
,
β
)
\left(\alpha ,\beta )
-higher
∗
\ast
-derivation (resp. a Jordan triple
(
α
,
β
)
\left(\alpha ,\beta )
-higher
∗
\ast
-derivation) of
R
R
if
d
n
(
a
2
)
=
∑
i
+
j
=
n
d
i
(
β
j
(
a
)
)
d
j
(
α
i
(
a
∗
i
)
)
{d}_{n}\left({a}^{2})={\sum }_{i+j=n}{d}_{i}\left({\beta }^{j}\left(a)){d}_{j}\left({\alpha }^{i}\left({a}^{{\ast }^{i}}))
(resp.
d
n
(
a
b
a
)
=
∑
i
+
j
+
k
=
n
d
i
(
β
j
+
k
(
a
)
)
d
j
(
β
k
(
α
i
(
b
∗
i
)
)
)
d
k
(
α
i
+
j
(
a
∗
i
+
j
)
)
{d}_{n}\left(aba)={\sum }_{i+j+k=n}{d}_{i}\left({\beta }^{j+k}\left(a)){d}_{j}\left({\beta }^{k}\left({\alpha }^{i}\left({b}^{{\ast }^{i}}))){d}_{k}\left({\alpha }^{i+j}\left({a}^{{\ast }^{i+j}}))
) for all
a
,
b
∈
R
a,b\in R
and each
n
∈
N
0
n\in {{\mathbb{N}}}_{0}
. We show that the two notions of Jordan
(
α
,
β
)
\left(\alpha ,\beta )
-higher
∗
\ast
-derivation and Jordan triple
(
α
,
β
)
\left(\alpha ,\beta )
-higher
∗
\ast
-derivation on a 6-torsion free semiprime
∗
\ast
-ring are equivalent.