Abstract
Let R be a prime ring, let
0
≠
b
∈
R
{0\neq b\in R}
, and let α and β be two automorphisms of R. Suppose that
F
:
R
→
R
{F:R\rightarrow R}
,
F
1
:
R
→
R
{F_{1}:R\rightarrow R}
are two b-generalized
(
α
,
β
)
{(\alpha,\beta)}
-derivations of R associated with the same
(
α
,
β
)
{(\alpha,\beta)}
-derivation
d
:
R
→
R
d:R\rightarrow R
, and let
G
:
R
→
R
G:R\rightarrow R
be a b-generalized
(
α
,
β
)
(\alpha,\beta)
-derivation of R associated with
(
α
,
β
)
(\alpha,\beta)
-derivation
g
:
R
→
R
g:R\rightarrow R
. The main objective of this paper is to investigate
the following algebraic identities:
(1)
F
(
x
y
)
+
α
(
x
y
)
+
α
(
y
x
)
=
0
{F(xy)+\alpha(xy)+\alpha(yx)=0}
,
(2)
F
(
x
y
)
+
G
(
x
)
α
(
y
)
+
α
(
y
x
)
=
0
{F(xy)+G(x)\alpha(y)+\alpha(yx)=0}
,
(3)
F
(
x
y
)
+
G
(
y
x
)
+
α
(
x
y
)
+
α
(
y
x
)
=
0
{F(xy)+G(yx)+\alpha(xy)+\alpha(yx)=0}
,
(4)
F
(
x
)
F
(
y
)
+
G
(
x
)
α
(
y
)
+
α
(
y
x
)
=
0
{F(x)F(y)+G(x)\alpha(y)+\alpha(yx)=0}
,
(5)
F
(
x
y
)
+
d
(
x
)
F
1
(
y
)
+
α
(
x
y
)
=
0
{F(xy)+d(x)F_{1}(y)+\alpha(xy)=0}
,
(6)
F
(
x
y
)
+
d
(
x
)
F
1
(
y
)
=
0
{F(xy)+d(x)F_{1}(y)=0}
,
(7)
F
(
x
y
)
+
d
(
x
)
F
1
(
y
)
+
α
(
y
x
)
=
0
{F(xy)+d(x)F_{1}(y)+\alpha(yx)=0}
,
(8)
F
(
x
y
)
+
d
(
x
)
F
1
(
y
)
+
α
(
x
y
)
+
α
(
y
x
)
=
0
{F(xy)+d(x)F_{1}(y)+\alpha(xy)+\alpha(yx)=0}
,
(9)
F
(
x
y
)
+
d
(
x
)
F
1
(
y
)
+
α
(
y
x
)
-
α
(
x
y
)
=
0
{F(xy)+d(x)F_{1}(y)+\alpha(yx)-\alpha(xy)=0}
,
(10)
[
F
(
x
)
,
x
]
α
,
β
=
0
{[F(x),x]_{\alpha,\beta}=0}
,
(11)
(
F
(
x
)
∘
x
)
α
,
β
=
0
{(F(x)\circ x)_{\alpha,\beta}=0}
,
(12)
F
(
[
x
,
y
]
)
=
[
x
,
y
]
α
,
β
{F([x,y])=[x,y]_{\alpha,\beta}}
,
(13)
F
(
x
∘
y
)
=
(
x
∘
y
)
α
,
β
{F(x\circ y)=(x\circ y)_{\alpha,\beta}}
for all
x
,
y
{x,y}
in some suitable subset of R.