Affiliation:
1. Department of Mathematics , Jadavpur University , Kolkata , 700032 , India
2. Department of Mathematics , Belda College , Belda , 721424 , India
Abstract
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.
Reference19 articles.
1. E. Albaş,
Generalized derivations on ideals of prime rings,
Miskolc Math. Notes 14 (2013), no. 1, 3–9.
2. A. Ali, V. De Filippis and F. Shujat,
On one sided ideals of a semiprime ring with generalized derivations,
Aequationes Math. 85 (2013), no. 3, 529–537.
3. F. Ali and M. A. Chaudhry,
On generalized
(
α
,
β
)
(\alpha,\beta)
-derivations of semiprime rings,
Turkish J. Math. 35 (2011), no. 3, 399–404.
4. M. Ashraf, A. Ali and S. Ali,
Some commutativity theorems for rings with generalized derivations,
Southeast Asian Bull. Math. 31 (2007), no. 3, 415–421.
5. M. Ashraf, A. Ali and R. Rani,
On generalized derivations of prime rings,
Southeast Asian Bull. Math. 29 (2005), no. 4, 669–675.