# Some Results on Generalizedα

Authors: Muhammad Anwar Chaudhry; Öznur Gölbaşi; Emine Koç
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

---

## Abstract

We extend some well known commutativity results concerning a nonzero square closed∗-Lie ideal and generalized α
,
β-derivations of ∗-prime rings.


---

## Body

## 1. Introduction

LetR be an associative ring with center Z. For any x
,
y
∈
R the symbol [
x
,
y
] represents commutator x
y
-
y
x. Recall that a ring R is prime if x
R
y
=
0 implies x
=
0 or y
=
0. An additive mapping ∗
:
R
→
R is called an involution if (
x
y
)
∗
=
y
∗
x
∗ and (
x
∗
)
∗
=
x for all x
,
y
∈
R. A ring equipped with an involution is called a ring with involution or ∗-ring. A ring with an involution is said to be ∗-prime if x
R
y
=
x
R
y
∗
=
0 or x
R
y
=
x
∗
R
y
=
0 implies that x
=
0 or y
=
0. Every prime ring with an involution is ∗-prime but the converse need not hold, in general. An example introduced by Oukhtite and Salhi [1] justifies the above statement; that is, R is a prime ring and S
=
R
×
R
o where R
o is the opposite ring of R. Define involution ∗ on S as ∗
(
x
,
y
)
=
(
y
,
x
). S is ∗-prime, but not prime. This example shows that ∗-prime rings constitute a more general class of prime rings. All that follows the symbol S
a
∗
(
R
), first introduced by Oukhtite and Salhi, will denote the set of symmetric and skew symmetric elements of R, that is, S
a
∗
(
R
)
=
{
x
∈
R
∣
x
∗
=
±
x
}.An additive subgroupU of R is said to be a Lie ideal of R if [
U
,
R
]
⊆
U. A Lie ideal is said to be a ∗-Lie ideal if U
∗
=
U. If U is a Lie ( ∗-Lie) ideal of R, then U is called a square closed Lie (∗-Lie) ideal of R if x
2
∈
U for all x
∈
U. An additive mapping d
:
R
→
R is called a derivation if d
(
x
y
)
=
d
(
x
)
y
+
x
d
(
y
) holds for all x
,
y
∈
R. For a fixed a
∈
R, the mapping I
a
:
R
→
R given by I
a
(
x
)
=
[
a
,
x
] is a derivation which is said to be an inner derivation. An additive mapping f
:
R
→
R is called a generalized derivation if there exists a derivation d
:
R
→
R such that (1)
f
x
y
=
f
x
y
+
x
d
y
,
∀
x
,
y
∈
R
.This definition was given by Bresar in [2].LetI be the ring of integers. Set R
=
a
b
0
c
∣
a
,
b
,
c
∈
I and U
=
0
b
0
0
∣
b
∈
I. We define the following maps on R
:
∗
a
b
0
c
=
c
-
b
0
a, f
a
b
0
c
=
a
2
b
0
0, and d
a
b
0
c
=
0
b
0
0. Then it is easy to see that U is a nonzero square closed ∗-Lie ideal of R and f is a generalized derivation associated with a nonzero derivation d commuting with ∗.Letα and β be any two automorphisms of R. An additive mapping d
:
R
→
R is called an (
α
,
β
)-derivation if d
(
x
y
)
=
d
(
x
)
α
(
y
)
+
β
(
x
)
d
(
y
) holds for all x
,
y
∈
R. Inspired by the definition of an (
α
,
β
)-derivation, the notion of a generalized derivation was extended as follows. Let α and β be any two automorphisms of R. An additive mapping f
:
R
→
R is called a generalized (
α
,
β
)-derivation on R if there exists an (
α
,
β
)-derivation d
:
R
→
R such that (2)
f
x
y
=
f
x
α
y
+
β
x
d
y
,
∀
x
,
y
∈
R
.Of course a generalized (
1,1
)-derivation is a generalized derivation on R, where 1 is the identity mapping on R.Several authors studied commutativity in prime and semiprime rings admitting derivations and generalized derivations which satisfy appropriate algebraic conditions on suitable subsets of the rings. Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from algebraic point of view might be interesting. Recently, some well-known results concerning prime rings have been proved for∗-prime ring by Oukhtite and Salhi (see [1], where further references can be found).In [3], Daif and Bell showed that the ideal I of a semiprime ring R is contained in the center of R if any of the following conditions is satisfied: (3)
d
x
,
y
=
x
,
y
,
d
x
,
y
=
-
x
,
y
∀
x
,
y
∈
I
.Several authors investigated this result for prime or semiprime ring admitting derivation or generalized derivation (see [4–6]). Also, Ashraf et al. proved some commutativity theorems for prime rings in [7, 8]. Motivated by these theorems, we will investigate some commutativity theorems for a nonzero square closed ∗-Lie ideal U and generalized (
α
,
β
)-derivations of ∗-prime rings.


## 2. Preliminaries

Throughout the paper,R will be a ∗-prime ring with characteristic not two and α
,
β
∈
A
u
t
R which commute with ∗. f will be a nonzero generalized α
,
β-derivation of R with associated nonzero α
,
β-derivation d which commutes with ∗ and U will be a nonzero ∗-Lie ideal of R. Also, we will make extensive use of the basic commutator identities:(4)
x
,
y
z
=
y
x
,
z
+
x
,
y
z
,
x
y
,
z
=
x
,
z
y
+
x
y
,
z
,
x
y
,
z
α
,
β
=
x
y
,
z
α
,
β
+
x
,
β
z
y
=
x
y
,
α
z
+
x
,
z
α
,
β
y
,
x
,
y
z
α
,
β
=
β
y
x
,
z
α
,
β
+
x
,
y
α
,
β
α
z
,
x
o
y
z
=
x
o
y
z
-
y
x
,
z
=
y
x
o
z
+
x
,
y
z
,
x
y
o
z
=
x
y
o
z
-
x
,
z
y
=
x
o
z
y
+
x
y
,
z
,
x
o
y
z
α
,
β
=
x
o
y
α
,
β
α
z
-
β
y
x
,
z
α
,
β
=
β
y
x
o
z
α
,
β
+
x
,
y
α
,
β
α
z
,
x
y
o
z
α
,
β
=
x
y
o
z
α
,
β
-
x
,
β
z
y
=
x
o
z
α
,
β
y
+
x
y
,
α
z
.Lemma 1 (see [1, Lemma 4]).
LetR be a ∗-prime ring with characteristic not two and let U be a nonzero ∗-Lie ideal of R and a, b
∈
R. If a
U
b
=
a
∗
U
b
=
0 or a
∗
U
b
=
a
U
b
=
0 then a
=
0 or b
=
0 or U
⊆
Z.Lemma 2 (see [9, Lemma 2.7]).
LetR be a ∗-prime ring with characteristic not two and let U be a nonzero ∗-Lie ideal of R. If a
∈
R such that [
a
,
U
]
⊆
Z, then either a
∈
Z or U
⊆
Z.The following lemma is an immediate consequence of Lemma2.Lemma 3.
LetR be a ∗-prime ring with characteristic not two and let U be a nonzero ∗-Lie ideal of R. If U
,
U
⊆
Z, then U
⊆
Z.Lemma 4 (see [10, Lemma 9]).
LetR be a ∗-prime ring with characteristic not two and let d be a nonzero α
,
β-derivation of R which commutes with ∗ and let U be a nonzero ∗-Lie ideal of R. If α, β commutes with ∗ and d
(
U
)
⊆
Z, then U
⊆
Z.Lemma 5 (see [10, Theorem 2]).
LetR be a ∗-prime ring with characteristic not two and let d be a nonzero α
,
β-derivation of R which commutes with ∗ and let U be a nonzero ∗-Lie ideal of R. If α, β commutes with ∗ and d
2
(
U
)
=
0, then U
⊆
Z.


## 3. Results

Theorem 6.
IfR satisfies any one of the following conditions,(i)
f
u
,
v
=
(
u
o
v
)
α
,
β, for all u
,
v
∈
U,
(ii)
f
u
,
v
=
-
(
u
o
v
)
α
,
β, for all u
,
v
∈
U,then U
⊆
Z.Proof.
(i) Iff
=
0, then (
u
o
v
)
α
,
β
=
0 for all u, v
∈
U. Replacing u by 2
u
w, w
∈
U, we get(5)
0
=
2
u
w
o
v
α
,
β
=
2
u
o
v
α
,
β
w
+
u
w
,
α
v
.Using c
h
a
r
R
≠
2 and the above equation in the last equation, we get(6)
u
w
,
α
v
=
0
,
∀
u
,
v
,
w
∈
U
;that is,(7)
U
w
,
α
v
=
0
,
∀
v
,
w
∈
U
.Hence we have(8)
u
U
w
,
α
v
=
0
,
∀
u
,
v
,
w
∈
U
.Since U is a nonzero ∗-Lie ideal of R, the above yields(9)
u
∗
U
w
,
α
v
=
0
,
∀
u
,
v
,
w
∈
U
.Therefore, we conclude that(10)
u
U
w
,
α
v
=
u
∗
U
w
,
α
v
=
0
,
∀
u
,
v
,
w
∈
U
.By Lemma 1, we get either [
w
,
α
(
v
)
]
=
0, for all v
,
w
∈
U or u
=
0, for all u
∈
U. And so, [
w
,
α
(
v
)
]
=
0, for all v
,
w
∈
U. By Lemma 2, we have α
(
v
)
∈
Z, for all v
∈
U or U
⊆
Z. Thus, we get U
⊆
Z.
We shall assume thatf
≠
0. We are given that(11)
f
u
,
v
=
u
o
v
α
,
β
,
∀
u
,
v
∈
U
.Replacing v by 2
v
u in (11) and using c
h
a
r
R
≠
2, we have(12)
f
u
,
v
u
=
u
o
v
α
,
β
α
u
-
β
v
u
,
u
α
,
β
.This implies that(13)
f
u
,
v
α
u
+
β
u
,
v
d
u
=
u
o
v
α
,
β
α
u
-
β
v
u
,
u
α
,
β
.By the hypothesis, we obtain(14)
β
u
,
v
d
u
=
-
β
v
u
,
u
α
,
β
,
∀
u
,
v
∈
U
.Substituting 2
v
w for v
,
w
∈
U in (14) and using c
h
a
r
R
≠
2, we get(15)
β
u
,
v
β
w
d
u
=
0
,
∀
u
,
v
,
w
∈
U
;that is,(16)
u
,
v
U
β
-
1
d
u
=
0
,
∀
u
,
v
∈
U
.Since U is a nonzero ∗-Lie ideal of R, we obtain(17)
u
,
v
∗
U
β
-
1
d
u
=
0
,
∀
v
∈
U
,
u
∈
U
∩
S
a
∗
R
;that is,(18)
u
,
v
U
β
-
1
d
u
=
u
,
v
∗
U
β
-
1
d
u
=
0
,
aaaaaaaaaaaaaaaaaa
∀
v
∈
U
,
u
∈
U
∩
S
a
∗
R
.By Lemma 1, we arrive at either u
,
v
=
0, for all v
∈
U or d
u
=
0 for each u
∈
U
∩
S
a
∗
R. Let u
∈
U, as u
+
u
∗, u
-
u
∗
∈
U
∩
S
a
∗
R, and u
±
u
∗
,
v
=
0, for all v
∈
U or d
u
±
u
∗
=
0. Hence, we have u
,
v
=
0 or d
u
=
0, for all u
,
v
∈
U. We set(19)
K
=
u
∈
U
∣
u
,
v
=
0
,
∀
v
∈
U
,
L
=
{
u
∈
U
∣
d
(
u
)
=
0
}
.Clearly each of K and L is additive subgroup of U. Moreover, U is the set-theoretic union of K and L. But a group cannot be the set-theoretic union of two proper subgroups; hence K
=
U or L
=
U. In the former case, U
,
U
=
0. By Lemma 3, we obtain U
⊆
Z. In the latter case, we have U
⊆
Z by Lemma 4. Thus the proof is completed.
(ii) Iff satisfies f
u
,
v
=
-
(
u
o
v
)
α
,
β, for all u
,
v
∈
U, then (
-
f
) satisfies the condition (
-
f
)
u
,
v
=
(
u
o
v
)
α
,
β and hence by part (i), our result follows.Theorem 7.
IfR satisfies any of the following conditions,(i)
f
u
o
v
=
[
u
,
v
]
α
,
β, for all u
,
v
∈
U,
(ii)
f
u
o
v
=
-
[
u
,
v
]
α
,
β, for all u
,
v
∈
U,then U
⊆
Z.Proof.
(i) Iff
=
0, then [
u
,
v
]
α
,
β
=
0, for all u
,
v
∈
U. Replacing u by 2
u
w, w
∈
U in the above equation and using c
h
a
r
R
≠
2, we obtain(20)
u
w
,
α
v
=
0
,
∀
u
,
v
,
w
∈
U
.Now repeating the similar arguments as we used in the proof of Theorem 6, we get the required result.
Henceforth, we assume thatf
≠
0. For any u
,
v
∈
U, we have(21)
f
u
o
v
=
u
,
v
α
,
β
.Replacing v by 2
v
u in (21) and using c
h
a
r
R
≠
2, we have(22)
f
u
o
v
u
=
u
,
v
α
,
β
α
u
+
β
v
u
,
u
α
,
β
;that is,(23)
f
u
o
v
α
u
+
β
u
o
v
d
u
=
u
,
v
α
,
β
α
u
+
β
v
u
,
u
α
,
β
.By application of (21), we find that(24)
β
u
o
v
d
u
=
β
v
u
,
u
α
,
β
,
∀
u
,
v
∈
U
.Taking 2
v
w for v
,
w
∈
U in (24) and using c
h
a
r
R
≠
2, (24), we get(25)
β
u
,
v
β
w
d
u
=
0
,
∀
u
,
v
,
w
∈
U
;that is,(26)
u
,
v
U
β
-
1
d
u
=
0
,
∀
u
,
v
∈
U
.The last expression is the same as (16) and hence the results follow.
(ii) Using similar techniques with necessary variations, we get the required result.Theorem 8.
Iff
(
u
)
,
v
α
,
β
=
(
f
(
u
)
o
v
)
α
,
β, for all u
,
v
∈
U, then U
⊆
Z.Proof.
We have(27)
f
(
u
)
,
v
α
,
β
=
f
u
o
v
α
,
β
,
∀
u
,
v
∈
U
.Replacing v by 2
v
w in (27) and using c
h
a
r
R
≠
2, we get(28)
f
u
,
v
α
,
β
α
w
+
β
v
f
u
,
w
α
,
β
=
f
u
o
v
α
,
β
α
(
w
)
-
β
(
v
)
f
(
u
)
,
w
α
,
β
.By the hypothesis, we obtain(29)
2
β
v
f
u
,
w
α
,
β
=
0
,
∀
u
,
v
,
w
∈
U
.Since c
h
a
r
R
≠
2 and β is an automorphism of R, we have(30)
U
β
-
1
f
u
,
w
α
,
β
=
0
,
∀
u
,
w
∈
U
.Hence we arrive at(31)
v
U
β
-
1
f
u
,
w
α
,
β
=
0
,
∀
u
,
v
,
w
∈
U
.Since U is a nonzero ∗-Lie ideal of R, the above yields(32)
v
∗
U
β
-
1
f
u
,
w
α
,
β
=
0
,
∀
u
,
v
,
w
∈
U
.Therefore, we get(33)
v
U
β
-
1
f
u
,
w
α
,
β
=
v
∗
U
β
-
1
f
u
,
w
α
,
β
=
0
,
∀
u
,
v
,
w
∈
U
.By Lemma 1, we get either f
(
u
)
,
w
α
,
β
=
0, for all u
,
w
∈
U or v
=
0, for all v
∈
U. Hence we conclude that(34)
f
(
u
)
,
w
α
,
β
=
0
,
∀
u
,
w
∈
U
.Substituting 2
u
v for u in (34) and using c
h
a
r
R
≠
2, (34), we get(35)
f
u
α
v
,
w
+
β
u
d
v
,
w
α
,
β
+
β
u
,
w
d
v
=
0
.Replacing w by v in the last equation, we get(36)
β
u
d
v
,
v
α
,
β
+
β
u
,
v
d
v
=
0
,
∀
u
,
v
∈
U
.Taking 2
u
w instead of u in the above equation and using this, we have(37)
β
u
,
v
β
w
d
u
=
0
,
∀
u
,
v
,
w
∈
U
;that is,(38)
u
,
v
U
β
-
1
d
u
=
0
,
∀
u
,
v
∈
U
.Using Lemma 1 and the similar arguments as we used after (16), we get the required result from (18).Theorem 9.
Iff
u
,
d
v
α
,
β
=
0, for all u
,
v
∈
U, then U
⊆
Z.Proof.
Substituting2
u
d
v, v
∈
[
U
,
U
] for u in the hypothesis and using c
h
a
r
R
≠
2, we obtain(39)
f
u
,
d
v
α
,
β
α
d
v
+
β
u
d
2
v
,
d
v
α
,
β
+
β
u
,
β
d
v
d
2
v
=
0
.Again using the hypothesis we have(40)
β
u
d
2
v
,
d
v
α
,
β
+
β
u
,
d
v
d
2
v
=
0
,
∀
u
∈
U
,
v
∈
[
U
,
U
]
.Replacing u by 2
u
w, w
∈
U in (40) and applying this equation, we arrive at(41)
2
β
u
,
d
v
β
w
d
2
v
=
0
,
∀
u
,
w
∈
U
,
v
∈
U
,
U
,so(42)
u
,
d
v
U
β
-
1
d
2
v
=
0
,
∀
u
∈
U
,
v
∈
[
U
,
U
]
.Since U is a nonzero square closed ∗-Lie ideal of R, we have [
U
,
U
] is a nonzero square closed ∗-Lie ideal of R, too. Hence using ∗
d
=
d
∗, we get(43)
u
,
d
v
∗
U
β
-
1
d
2
v
=
0
,
∀
u
∈
U
,
v
∈
[
U
,
U
]
∩
S
a
∗
(
R
)
.By the application of Lemma 1, we get either u
,
d
v
=
0, for all u
∈
U or β
-
1
d
2
v
=
0, for each v
∈
[
U
,
U
]
∩
S
a
∗
(
R
). Since β is an automorphism of R, we have either u
,
d
v
=
0, for all u
∈
U or d
2
v
=
0, for each v
∈
[
U
,
U
]
∩
S
a
∗
(
R
). Let v
∈
[
U
,
U
], as v
+
v
∗, v
-
v
∗
∈
[
U
,
U
]
∩
S
a
∗
(
R
) and [
u
,
d
(
v
∓
v
∗
)
]
=
0, for all u
∈
U or d
2
(
v
∓
v
∗
)
=
0. Hence, we have u
,
d
v
=
0 or d
2
v
=
0, for all u
∈
U, v
∈
[
U
,
U
].
Now, we set(44)
K
=
v
∈
U
,
U
∣
u
,
d
v
=
0
,
∀
u
∈
U
,
L
=
v
∈
U
,
U
∣
d
2
v
=
0
.Clearly each of K and L is additive subgroup of [
U
,
U
]. Moreover, [
U
,
U
] is the set-theoretic union of K and L. But a group cannot be the set-theoretic union of two proper subgroups; hence K
=
[
U
,
U
] or L
=
[
U
,
U
]. In the former case, we have U
,
d
v
=
0 for all v
∈
U
,
U. Using Lemma 2, we get d
v
∈
Z for all v
∈
U
,
U or U
⊆
Z. By Lemmas 4 and 3, we have U
⊆
Z. In the latter case, d
2
(
[
U
,
U
]
)
=
0; that is, [
U
,
U
]
⊆
Z by Lemma 5, so again using Lemma 3, we get U
⊆
Z. This completes the proof.Theorem 10.
Iff
u
,
d
v
α
,
β
=
v
α
u, for all u
,
v
∈
U, then U
⊆
Z.Proof.
Iff
=
0, then v
α
(
u
)
=
0, for all u
,
v
∈
U. Replacing v by 2
v
w in this equation and using c
h
a
r
R
≠
2, we see that(45)
v
U
α
u
=
0
,
∀
u
,
v
∈
U
.Since U is a nonzero ∗-Lie ideal of R, we get(46)
v
∗
U
α
u
=
0
,
∀
u
,
v
∈
U
.By Lemma 1, we have v
=
0 or u
=
0, for all u, v
∈
U, and so U
⊆
Z.
Now, we assumef
≠
0. By the hyphothesis, we get(47)
f
u
,
d
v
α
,
β
=
v
α
u
,
∀
u
,
v
∈
U
.Replacing u by 2
u
w, w
∈
U in (47) and using our hypothesis, we get(48)
2
f
(
u
)
[
α
(
w
)
,
α
(
d
(
v
)
)
]
+
[
β
(
u
)
,
β
(
d
(
v
)
)
]
d
(
w
)
d
w
,
d
v
α
,
β
+
β
(
u
)
d
w
,
d
v
α
,
β
=
0
.Again replacing w by d
v, v
∈
[
U
,
U
] in the last equation and using c
h
a
r
R
≠
2, we obtain(49)
β
u
,
β
d
v
d
2
v
+
β
u
d
2
v
,
d
v
α
,
β
=
0
,
∀
u
∈
U
,
v
∈
[
U
,
U
]
.Taking 2
u
w instead of u in (49) and using this equation, we have(50)
2
β
u
,
d
v
β
w
d
2
v
=
0
,
∀
u
,
w
∈
U
,
v
∈
[
U
,
U
]
,so(51)
u
,
d
v
U
β
-
1
d
2
v
=
0
,
∀
u
∈
U
,
v
∈
[
U
,
U
]
.By further application of similar arguments as used in the end of proof of Theorem 9, we get the required result.Theorem 11.
IfR satisfies any of the following conditions,(i)
d
(
u
)
f
(
v
)
=
[
u
,
v
]
α
,
β, for all u
,
v
∈
U,
(ii)
d
(
u
)
f
(
v
)
=
-
[
u
,
v
]
α
,
β, for all u
,
v
∈
U,then U
⊆
Z.Proof.
Iff
=
0, then [
u
,
v
]
α
,
β
=
0, for all u, v
∈
U. Now using the similar arguments as used in the proof of Theorem 7, we get the required result.
Suppose thatf
≠
0 and(52)
d
(
u
)
f
(
v
)
=
u
,
v
α
,
β
,
∀
u
,
v
∈
U
.Replacing v by 2
v
w, w
∈
U and using the fact that c
h
a
r
R
≠
2, we have(53)
d
u
β
v
d
w
=
β
v
u
,
w
α
,
β
,
∀
u
,
v
,
w
∈
U
.Replacing 2
v
t, t
∈
U for v in (53) we get(54)
d
u
β
v
β
t
d
w
=
β
v
β
t
u
,
w
α
,
β
,
∀
u
,
v
,
w
,
t
∈
U
.Using (49) in the last equation, we get(55)
d
u
β
v
β
t
d
w
=
β
v
d
u
β
t
d
w
,
∀
u
,
v
,
w
,
t
∈
U
,so(56)
d
u
,
β
v
β
t
d
w
=
0
,
∀
u
,
v
,
w
,
t
∈
U
.Since β is automorphism of R, we obtain(57)
β
-
1
d
u
,
v
U
β
-
1
d
w
=
0
,
∀
u
,
v
,
w
∈
U
.Using β
∗
=
∗
β, ∗
d
=
d
∗ and U is a nonzero ∗-Lie ideal of R, we have(58)
β
-
1
d
u
,
v
U
β
-
1
d
w
∗
=
0
,
∀
u
,
v
,
w
∈
U
.By Lemma 1, we get either β
-
1
d
u
,
v
=
0 or β
-
1
d
w
=
0, for all u, v, w
∈
U. If β
-
1
d
u
,
v
=
0, for all u
,
v
∈
U, then we have d
U
⊆
Z by Lemma 2. Hence we conclude that d
U
⊆
Z or d
U
=
0, so d
U
⊆
Z in both cases; thus we get U
⊆
Z by Lemma 4.Theorem 12.
Iff
u
v
-
u
α
v
∈
C
α
,
β, for all u
,
v
∈
U, then U
⊆
Z.Proof.
Iff
=
0, then u
α
(
v
)
∈
C
α
,
β, for all u
,
v
∈
U. In particular, [
u
α
(
v
)
,
v
]
α
,
β
=
0, for all u
,
v
∈
U and hence(59)
u
,
v
α
,
β
α
(
v
)
=
0
,
∀
u
,
v
∈
U
.Replacing u by 2
u
w, we get 2
[
u
,
β
(
v
)
]
w
α
(
v
)
=
0, for all u, v, w
∈
U. Since c
h
a
r
R
≠
2, we see that(60)
u
,
β
v
U
α
v
=
0
,
∀
u
,
v
∈
U
;that is,(61)
α
-
1
u
,
β
v
U
v
=
0
,
∀
u
,
v
∈
U
.Since U is a nonzero ∗-Lie ideal of R, we get α
-
1
(
[
u
,
β
(
v
)
]
)
U
v
∗
=
0, for all u
∈
U, v
∈
U
∩
S
a
∗
(
R
). By Lemma 1, we get [
u
,
β
(
v
)
]
=
0 or v
=
0 for all u
∈
U, v
∈
U
∩
S
a
∗
(
R
). This implies that [
U
,
β
(
v
)
]
=
0, for all v
∈
U
∩
S
a
∗
(
R
). Let v
∈
U, as v
+
v
∗, v
-
v
∗
∈
U
∩
S
a
∗
(
R
), and [
U
,
β
(
v
±
v
∗
)
]
=
0. Hence we have [
U
,
β
(
v
)
]
=
0, for all v
∈
U. By Lemma 2, we find that β
(
v
)
∈
Z. Thus, we get U
⊆
Z.
Hence, onward we assume thatf
≠
0. By the hypothesis, we have(62)
f
u
v
-
u
σ
v
∈
C
α
,
β
,
∀
u
,
v
∈
U
.This can be rewritten as(63)
f
u
α
v
+
β
u
d
v
-
u
α
v
∈
C
α
,
β
,
∀
u
,
v
∈
U
.Replacing v by 2
v
w
,
w
∈
U in (63) and using c
h
a
r
R
≠
2, we arrive at(64)
f
u
α
v
+
β
u
d
v
-
u
α
v
α
w
+
β
u
β
v
d
w
∈
C
α
,
β
,
∀
u
,
v
,
w
∈
U
.Commuting this term with w and using the hypothesis, we obtain(65)
β
u
v
d
w
,
w
α
,
β
=
0
,
∀
u
,
v
,
w
∈
U
,so(66)
β
u
v
d
w
,
w
α
,
β
+
β
u
v
,
β
w
d
w
=
0
,
∀
u
,
v
,
w
∈
U
.Replacing 2
v
t, t
∈
U by v in this equation and using this, we get(67)
β
(
u
)
,
β
w
β
2
v
t
d
w
=
0
,
∀
u
,
v
,
w
,
t
∈
U
,so(68)
u
,
w
U
2
β
-
1
d
w
=
0
,
∀
u
,
w
∈
U
.By trivial modification of the argument in the proof of Theorem 8, we get the required result.Theorem 13.
Iff
u
v
+
u
α
v
∈
C
α
,
β for all u
,
v
∈
U, then U
⊆
Z.Proof.
Iff is a generalized α
,
β-derivation satisfying the property f
u
v
+
u
α
v
∈
C
α
,
β for all u
,
v
∈
U, then -
f satisfies the condition -
f
u
v
-
u
α
v
∈
C
α
,
β, for all u
,
v
∈
U. Hence we have U
⊆
Z by Theorem 12.



---

*Source: 583050-2015-04-28.xml*