Journal of Mathematical Extension
Vol. 10, No. 3, (2016), 119-135
ISSN: 1735-8299
Journal of Mathematical
Extension
of Mathematical
Extension
URL: Journal
http://www.ijmex.com
Vol.Vol.
10, 10,
No.No.
3, (2016),
119-135
3, (2016),
119-135
ISSN:
1735-8299
ISSN:
1735-8299
URL:
http://www.ijmex.com
URL:
http://www.ijmex.com
A General Characterization of Additive
Maps on Semiprime Rings
AA
General
Characterization
of of
Additive
General
Characterization
Additive
A.
Hosseini
Maps
onon
Semiprime
Rings
Maps
Semiprime
Rings
Kashmar Higher Education Institute
A. A.
Hosseini
Hosseini
Kashmar
Higher
Education
Institute
RKashmar
be a 2-torsion
freeEducation
semiprime
ring
and T : R → R be
Abstract. Let
Higher
Institute
1.
a Jordan left centralizer associated with a lsemi Hochschild 2cocycle
: RR → R. Then, T is a left centralizer associated with . Applying
Let Let
R be
2-torsion
free free
semiprime
ringring
andand
T :R
Abstract.
Raprove
be
a 2-torsion
semiprime
T :→
RR
→be
R
Abstract.
this
main result,
we
that every
Jordan
generalized
derivation
onbe
a Jordan
left left
centralizer
associated
withwith
a lsemi
Hochschild
2cocycle
a
Jordan
centralizer
associated
a
lsemi
Hochschild
2cocycle
a 2-torsion free semiprime ring is a generalized derivation.
: RR
→ R.
T isTa is
left
centralizer
associated
withwith
. Applying
: RR
→ Then,
R. Then,
a left
centralizer
associated
. Applying
this
main
result,
we
prove
thatthat
every
Jordan
generalized
on on
this
main
result,
we prove
every
Jordan
generalized
derivation
AMS
Subject
Classication:
47B47;
17B40;
16N60 derivation
a Keywords
2-torsion
free
semiprime
ring
is a isgeneralized
derivation.
a 2-torsion
free
semiprime
ring
a generalized
derivation.
and
Phrases:
Jordan
derivation,
jordan
centralizer, l-semi
hochschild 2-cocycle, 2-torsion free semiprime ring
AMS
Subject
Classication:
47B47;
17B40;
16N60
AMS
Subject
Classication:
47B47;
17B40;
16N60
Keywords
and
Phrases:
Jordan
derivation,
jordan
centralizer,
l-semi
Keywords and Phrases: Jordan derivation, jordan
centralizer,
l-semi
hochschild
2-cocycle,
2-torsion
freefree
semiprime
ringring
hochschild
2-cocycle,
2-torsion
semiprime
Introduction
this paper, R denotes an associative ring with the center
1.Throughout
1. Introduction
Introduction
Z(R), T is considered as an additive map on R and is a biadditive
map
from Rthis
× this
Rpaper,
into
R.
anan
integer
n 2, ring
a ring
Rwith
isthe
said
tocenter
be
Throughout
R Given
denotes
associative
with
center
Throughout
paper,
R denotes
an
associative
ring
the
n-torsion
free,
if
for
x
∈
R,
nx
=
0
implies
that
x
=
0.
We
denote
by
Z(R),
T
is
considered
as
an
additive
map
on
R
and
is
a
biadditive
Z(R), T is considered as an additive map on R and is a biadditive
map
from
Rcommutator
×RR×into
[x,map
y],
the
Given
yx,an
foran
allinteger
x, yn∈nR.
that
aisring
R.xy
Given
integer
2,Recall
a2,ring
R
isRsaid
to R
be
from
R into
R.
a ring
said
toisbe
prime
if
for
x,
y
∈
R,
n-torsion
free,
if
for
xRy
=
{0}
implies
that
either
x
=
0
or
y
=
0,
and
x
∈
R,
nx
=
0
implies
that
x
=
0.
We
denote
by
n-torsion free, if for x ∈ R, nx = 0 implies that
0. We denote by
is y],
semiprime
in the case
that
xthat
= 0.
[x,
thethe
commutator
xythat
for =
all{0}
x, yimplies
Recall
aAn
ring
R isR is
[x,
y],
commutator
xyyx,
xRx
yx,
for
all
x,∈yR.
∈ R.
Recall
that
aadditive
ring
prime
if
for
x,
y
∈
R,
map
D
:
R
→
R
is
called
a
derivation
if
D(xy)
=
D(x)y
+
xD(y)
xRy
=
{0}
implies
that
either
x
=
0
or
y
=
0,
and
prime if for x, y ∈ R, xRy = {0} implies that either x = 0 or y =holds
0, and
2) =
isfor
semiprime
case
thatthat
x, y ∈inRthe
and
is case
called
a Jordan
in the
D(x
xRx
= {0}
implies
that
x case
=x0.=that
An
additive
isall
semiprime
in
the
xRx
=derivation
{0}
implies
that
0. An
additive
map
D(x)x
is called
fulfilled
for
all x ∈ifR.
every
derivation
is
D :+
a derivation
D(xy)
= D(x)y
+ xD(y)
holds
map
DRxD(x)
:→
RR
→ isR
is called
a derivation
ifObviously,
D(xy)
= D(x)y
+
xD(y)
holds
2
2
a for
Jordan
the
is not in
true,
in general.
AD(x
wellfor
all
x,
is called
a Jordan
derivation
the
case
thatthat
D(x
) =) =
allyx,∈derivation,
yR∈and
R and
isbut
called
aconverse
Jordan
derivation
in the
case
known
ofisHerstein
[8] for
states
in Obviously,
the caseevery
that
D(x)x
+result
xD(x)
fulfilled
for
all all
x ∈xthat
R.
Obviously,
derivation
is is
R derivation
is a prime
D(x)x
+ xD(x)
is fulfilled
∈ R.
every
a Jordan
derivation,
but
the
converse
is
not
true,
in
general.
A
wella Received:
Jordan December
derivation,
but the converse
is not true, in general. A well2015; Accepted: March 2016
2015
known
result
of Herstein
[8] [8]
states
thatthat
in the
casecase
thatthat
R is
known
result
of Herstein
states
in the
R aisprime
a prime
119
Received: December 2015; Accepted: March 2015
Received: December 2015; Accepted: March 2015
119119
120
A. HOSEINI
ring of characteristic not 2, then every Jordan derivation D : R → R is a
derivation. A brief proof of this result has been presented in [5]. Cusack
[6] has extended Herstein’s result to 2-torsion free semiprime rings (see
also [4] for an alternative proof). An additive map T : R → R is called
a left (resp. right) centralizer if T (xy) = T (x)y (resp. T (xy) = xT (y))
holds for all x, y ∈ R. We call T a centralizer whenever T is both a left
and a right centralizer. An additive map T : R → R is called a Jordan
left (right) centralizer when T (x2 ) = T (x)x (resp. T (x2 ) = xT (x)) holds
for all x ∈ R. Following some ideas from [4], Zalar [11] proved that any
left (resp. right) Jordan centralizer on a 2-torsion free semiprime ring
is a left (resp. right) centralizer. By using the main results of [4] and
[11], Vukman [10] proved that every Jordan generalized derivation on a
2-torsion free semiprime ring is a generalized derivation. By using the
main result of this paper, we offer an alternative proof for this result of
Vukman.
A bi-additive map : R × R → R is said to be a l-semi Hochschild
2-cocycle if
(xy, z) (x, yz) + (x, y)z = 0,
for all x, y, z ∈ R. A l-semi Hochschild 2-cocycle is said to be symmetric (resp. anti symmetric) if (x, y) = (y, x) (resp. (x, y) = (y, x))
for all x, y ∈ R. We say that an additive map T : R → R is a left centralizer associated with , if there exists a bi-additive map : R × R → R
such that T (xy) = T (x)y + (x, y) holds for all x, y ∈ R. Clearly, in
this case we have
(xy, z) (x, yz) + (x, y)z = T (xyz) T (xy)z T (xyz)+
T (x)yz + T (xy)z T (x)yz = 0,
for all x, y, z ∈ R. It means that is a l-semi Hochschild 2-cocycle. Let
: R × R → R be a l-semi Hochschild 2-cocycle. An additive map
T : R → R is said to be a Jordan left centralizer associated with ,
whenever T (x2 ) = T (x)x + (x, x) for all x ∈ R. A bi-additive map
λ : R × R → R is called a r-semi Hochschild 2-cocycle if λ(z, xy)
λ(zx, y) + zλ(x, y) = 0 for all x, y, z ∈ R. An additive map T : R → R
A
A GENERAL
GENERAL CHARACTERIZATION
CHARACTERIZATIONOF
OFADDITIVE
ADDITIVE......
121
121
is
is said
said to
to be
be aa right
right centralizer
centralizer associated
associated with
withλ,λ,ififthere
thereexists
existsaabibiadditive
additive map
map λλ :: R
R×
×R
R→
→R
R such
such that
thatTT(xy)
(xy)==xT
xT(y)
(y)++λ(x,
λ(x,y)y)for
forall
all
x,
x, yy ∈
∈ R.
R. Obviously,
Obviously, λλ isis aa r-semi
r-semi Hochschild
Hochschild2-cocycle.
2-cocycle.
Let
Let TT(x
(x22)) =
= TT(x)x
(x)x +
+(x,
(x,x)
x) for
for all
all xx ∈∈RR (1).
(1).Replacing
Replacingxxby
byxx++yy
in
in (1),
(1), we
we get
get TT(xy
(xy +
+yx)
yx) =
= TT(x)y
(x)y++TT(y)x
(y)x++(x,
(x,y)
y)++(y,
(y,x)x)for
forall
all
x,
x, yy ∈
∈R
R (2).
(2). Note
Note that
that ifif R
R isis aa 2-torsion
2-torsionfree
freering,
ring,then
then(1)
(1)and
and(2)
(2)
are
are equivalent.
equivalent. Some
Some authors
authors define
defineaaJordan
Jordanleft
leftcentralizer
centralizerasasfollows:
follows:
An
An additive
additive map
map TT :: R
R→
→R
R isis called
called aa Jordan
Jordan left
left centralizer
centralizerifif
TT(xy
(xy +
+ yx)
yx) =
= TT(x)y
(x)y +
+ TT(y)x
(y)x holds
holds for
for all
all x,x,yy ∈∈ R.
R. With
Withthis
thishyhypothesis,
pothesis, we
we have
have TT(xy)
(xy)
TT(x)y
(x)y == (T
(T(yx)
(yx)TT(y)x)
(y)x) (3).
(3).IfIfwe
wededefine
fine (x,
(x,y)
y) =
= TT(xy)
(xy)
TT(x)y
(x)y (x,
(x,yy ∈∈ R),
R), then
thenititfollows
followsfrom
from(3)
(3)that
that
(x,
(x,y)
y) =
= (y,
(y,x)
x) and
and itit means
means that
that isis anti
anti symmetric.
symmetric. Suppose
Suppose
that
that TT :: R
R→
→R
R isis aa left
left centralizer
centralizer associated
associated with
with,,i.e.
i.e.TT(xy)
(xy)==
TT(x)y
(x)y +
+ (x,
(x,y)
y) for
for all
all x,
x,yy ∈∈ R.
R. IfIf isis anti
anti symmetric,
symmetric,then
thenwe
wehave
have
TT(xy)
(xy)
TT(x)y
(x)y =
= T
T(yx)
(yx) +
+ TT(y)x
(y)x and
and consequently,
consequently, TT(xy
(xy++yx)
yx) ==
TT(x)y
(x)y +
+TT(y)x
(y)x for
for all
all x,
x,yy ∈∈R.
R.ItItmeans
meansthat
thatTT isisaaJordan
Jordanleft
leftcentralcentralizer.
izer. Therefore,
Therefore, IfIf :: R
R×
×R
R→
→R
R defined
definedby
by(x,
(x,y)
y)==TT(xy)
(xy)TT(x)y
(x)y
is
is anti
anti symmetric,
symmetric, then
then TT isis aa Jordan
Jordanleft
leftcentralizer
centralizerififand
andonly
onlyififTTisis
aa left
left centralizer
centralizer associated
associated with
with the
thel-semi
l-semiHochschild
Hochschild2-cocycle
2-cocycle..
Similar
Similar to
to the
the approach
approach presented
presented in
in [11],
[11],we
weprove
provethe
thefollowing
followingmain
main
result:
result:
Let
Let R
R be
be aa 2-torsion
2-torsion free
free semiprime
semiprime ring
ring and
andTT : :RR→
→RRbe
beaaJordan
Jordan
left
left centralizer
centralizer associated
associated with
with ,
, where
where isis aa l-semi
l-semi Hochschild
Hochschild2-2cocycle.
cocycle. Then,
Then, TT isis aa left
left centralizer
centralizerassociated
associatedwith
with.
.
In
In this
this paper,
paper, we
we show
show that
that derivations,
derivations, generalized
generalized derivations,
derivations, σ-σderivations,
derivations, generalized
generalizedσ-derivations,
σ-derivations,(σ,
(σ,ττ)-derivations
)-derivationsand
andθ-centralizers
θ-centralizers
are
are left
left centralizer
centralizer associated
associatedwith
withaasuitable
suitablel-semi
l-semiHochschild
Hochschild2-cocycle.
2-cocycle.
This
This means
means that
that the
the aforementioned
aforementioned concepts
concepts can
can be
be unified
unifiedand
andinintegrated
tegrated together.
together. By
By reviewing
reviewing some
some papers
papers (( [1],
[1], [2],
[2], [3],
[3],and
andreferreferences
ences therein)
therein) about
about Jordan
Jordan left
left(θ,
(θ,φ)-derivations,
φ)-derivations,(θ,
(θ,φ)-derivations,
φ)-derivations,τ τ- centralizers
centralizers and
and -centralizers
-centralizers on
on 2-torsion
2-torsion free
free semiprime
semiprimerings,
rings,ititisis
observed
observed that
that the
the maps
maps like
like θ,θ, φ,
φ, and
and ττ are
are supposed
supposed asas homomorhomomorphism
phism and
and automorphism.
automorphism. We
We believe
believe this
this assumption
assumptioncan
canreduce
reducethe
the
generality
generality of
of the
the topic.
topic. In
In this
this paper,
paper,therefore,
therefore,we
weare
aregoing
goingtotopresent
present
122
A. HOSEINI
some results about Jordan σ-derivations, Jordan (σ, τ )-derivations, Jordan generalized derivations and Jordan generalized σ-derivations based
on the new type of left centralizers, while τ , σ and θ are not supposed homomorphism, necessarily. It is one of the reasons that obviously proves
performance and application of this type of centralizers.
2.
Centralizer Associated with Semi Hochschild
2-Cocycle
Definition 2.1. A biadditive map : R × R → R is said to be a lsemi Hochschild 2-cocycle if (xy, z) (x, yz) + (x, y)z = 0 for all
x, y, z ∈ R.
Definition 2.2. For a biadditive map : R × R → R, an additive
map T : R → R is said to be a left centralizer associated with if
T (xy) = T (x)y + (x, y) for all x, y ∈ R.
As we mentioned in the introduction, such is a l-semi Hochschild 2cocycle.
Example 2.3. Every derivation D : R → R is a left centralizer associated with : R×R → R defined by (x, y) = xD(y) for all x, y ∈ R. We
have
(xy, z) (x, yz) + (x, y)z = xyD(z) xD(yz) + xD(y)z
= xyD(z) xyD(z) xD(y)z + xD(y)z
= 0.
Hence, D(xy) = D(x)y + xD(y) = D(x)y + (x, y) is a left centralizer
associated with .
Example 2.4. Suppose D : R → R is a derivation. Then, every Dderivation f : R → R is a left centralizer associated with , if is
defined as above.
Example 2.5. Suppose θ : R → R is an endomorphism. Then, h =
θ id, where id is the identity mapping on R , is a (θ, id) derivation
A GENERAL
GENERAL CHARACTERIZATION
CHARACTERIZATIONOF
OFADDITIVE
ADDITIVE......
123
123
as following:
following:
h(xy) = θ(xy)
θ(xy)
xy
xy =
= θ(x)θ(y)
θ(x)θ(y)θ(x)y
θ(x)y++θ(x)y
θ(x)yxy
xy==h(x)y
h(x)y++θ(x)h(y).
θ(x)h(y).
If we define
define (x,
(x,y)
y) =
= θ(x)h(y)
θ(x)h(y) for
for all
all x,x,yy ∈∈ R,
R,then
thenisisa al-semi
l-semi
Hochschild
Hochschild 2-cocycle.
2-cocycle. It
It means
meansthat
thathhisisaaleft
leftcentralizer
centralizerassociated
associatedwith
with
.
Furthermore,
Furthermore, suppose
suppose that
that TT ::R
R→
→RRisisaaleft
leftθθcentralizer,
centralizer,i.e.
i.e.T T
is additive
additive and
and TT(xy)
(xy) =
= TT(x)θ(y)
(x)θ(y) holds
holdsfor
forall
allx,x,yy∈∈R.
R.Considering
Considering
h = θ id,
id, we
we have
have
TT (xy)
(xy) =
= TT(x)θ(y)
(x)θ(y)==TT(x)(h
(x)(h++id)(y)
id)(y)==TT(x)(h(y)
(x)(h(y)++y)y)
=
= TT(x)y
(x)y+
+TT(x)h(y).
(x)h(y).
Defining (x,
(x,y)
y) =
= TT(x)h(y)
(x)h(y) for
for all
all x,x,yy ∈∈ R,
R,we
weconclude
concludethat
thatisis
a l-semi Hochschild
Hochschild 2-cocycle.
2-cocycle. Hence,
Hence, TT isisaaleft
leftcentralizer
centralizerassociated
associated
with .
Example
Example 2.6.
2.6. Let
Let σ,
σ,ττ :: R
R→
→R
Rbe
betwo
twoendomorphisms
endomorphismsand
andd d: R
: R→→RR
be a σ-derivation.
σ-derivation.
(i) Every (σ,
(σ, ττ)-derivation
)-derivation FF :: R
R→
→ RR isisaaleft
leftcentralizer
centralizerassociated
associated
with , if
is
is defined
defined by
by (x,
(x,y)
y)==FF(x)(σ
(x)(σid)(y)
id)(y)++τ τ(x)F
(x)F(y).
(y).
(ii) Suppose
Suppose δδ :: R
R→
→R
R isis aa generalized
generalizedσ-derivation.
σ-derivation. Put
Put(x,
(x,y)y)==
δ(x)(σ id)(y)
id)(y) +
+ σ(x)d(y),
σ(x)d(y), where
wheredd: :RR→
→RRisisaaσ-derivation,
σ-derivation,sosoδ δisis
a left centralizer
centralizer associated
associated with
with .
. Let
LetTT : :RR→
→RRbebea aJordan
Jordanleft
left
centralizer
centralizer associated
associated with
with ,
, and
anddefine
defineψψ: :RR××RR→→RRbybyψ(x,
ψ(x,y)y)==
T (xy) TT(x)y
(x)y
(x,
(x,y)
y) for
for all
all x,
x,yy ∈∈ R.
R. The
Thefollowing
followingproposition
proposition
demonstrates
demonstrates several
several properties
propertiesofofψ.
ψ.
Proposition
Proposition 2.7.
2.7. Let
Let ψ,
ψ, and
and TT bebeas
asabove.
above. The
Thebi-additive
bi-additivemap
mapψ ψ
satisfies the
the following:
following:
(1) ψ is anti
anti symmetric,
symmetric, i.e.
i.e. ψ(x,
ψ(x,y)
y)==ψ(y,
ψ(y,x),
x),
(2) ψ is aa l-semi
l-semi Hochschild
Hochschild 2-cocycle,
2-cocycle,
(3) 2ψ(xy,
2ψ(xy, z)
z) +
+ ψ(x,
ψ(x,y)z
y)z
ψ(y,
ψ(y,z)x
z)x++ψ(z,
ψ(z,x)y
x)y==0,0,
(4) If R is
is aa 2-torsion
2-torsion free
free ring,
ring, then
thenψ(xy,
ψ(xy,z)z)++ψ(xz,
ψ(xz,y)y)==0,0,
(5) If R is
is aa 2-torsion
2-torsion free
free ring,
ring, then
thenψ([x,
ψ([x,y],
y],z)z)==ψ(x,
ψ(x,y)z,
y)z,
(6) If R is
is aa 2-torsion
2-torsion free
free ring,
ring, then
thenψ(x,
ψ(x,y)[z,
y)[z,w]
w]++ψ(z,
ψ(z,w)[x,
w)[x,y]y]==0,0,
124
124
124
A.A.
HOSEINI
A.
HOSEINI
HOSEINI
(7)
ψ([x,
y]z,
w)w)
==
0,
(7)
R isisaa2-torsion
2-torsionfree
freering,
ring,then
then
ψ([x,
y]z,
w)
=0,0,
(7) IfIf
If R
2-torsion
free
ring,
then
ψ([x,
y]z,
(8)
ψ(x,
y)[z,
w]r
==
0,
(8)
R isisaa2-torsion
2-torsionfree
freering,
ring,then
then
ψ(x,
y)[z,
w]r
=0,0,
(8) IfIf
If R
2-torsion
free
ring,
then
ψ(x,
y)[z,
w]r
for
for
all
x,y,y,z,z,rr∈∈R.
for all
all x,
R.
Proof.
Proof.
(1)and
and(2)
arestraightforward.
straightforward.
Proof.(1)
(2)are
are
straightforward.
For
proving
(3),
we
have
For
proving
(3),
we
have
For proving
have
2ψ(xy,
z)x
++
ψ(z,
x)y
===
2ψ(xy,
z)z)
+
ψ(x,
y)zy)z
++
ψ(z,
y)xy)x
2ψ(xy,
z)++ψ(x,
ψ(x,y)z
ψ(y,
ψ(y,
z)x
+
ψ(z,
x)y
2ψ(xy,
z)+
+ψ(x,
ψ(x,
y)z
+ψ(z,
ψ(z,
y)x
2ψ(xy,z)
y)z
ψ(y,
z)x
ψ(z,
x)y
2ψ(xy,
+
(xy,
z)z)
(xy,
z)z)
T(x)yz
(x,
y)zy)z
+
ψ(z,
x)y==TT(xyz)
TTT(xy)z
(xy)z
(xy,
z)
(xy,
(xy,
z)
(x)yz
(x,
(x,
y)z
+ψ(z,
ψ(z,x)y
(xyz)
(xy)z
(xy,
TT(x)yz
+
y)x
++
TT(zx)y
T
(z)xy
(z,
x)y
++
T+
(xyz)
+
(zy)xTT(z)yx
(z,
y)x
+
T(zx)y
(zx)y
T
T(z)xy
(z)xy
(z,
(z,
x)y
(xyz)
+TTT(zy)x
(z)yx(z,
(z,
y)x
x)y
TT(xyz)
=
xy)
(xy,
z)z)
++T+(yz)x
T(yzx)
++
(yz,
x) x)x)
=
T
(zxy)++TT(z)xy
+(z,
(z,
xy)
(xy,
z)
(yz)x
(yzx)
+(yz,
(yz,
=T
T(zxy)
(z)xy++
(z,
xy)
(xy,
TT(yz)x
TT(yzx)
+
(zy)x
T T(z)yx
(z,
y)x
++
T+(zx)y
T
(z)xy
+
(x,
yz)(x,
+TTT
(zy)x
T(z)yx
(z)yx
(z,
(z,
y)x
(zx)y
(z)xy
+(x,
(x,yz)
(x,y)z
y)z++
(zy)x
y)x
TT(zx)y
TT(z)xy
(y,
zx)
(zx,
y)y)
+
(z,
xy)
(xy,
z) z)z)
(z,
x)y==TT(yzx)
TTT(y)zx
(y)zx
(y,
zx)
(zx,
(zx,
y)+
+(z,
(z,
xy)
(xy,
(xy,
(z,
(z,x)y
(yzx)
(y)zx
(y,
zx)
xy)
+
x)x)
++
(x,
yz)
(x,
y)z
++
T+(zy)x
T
(z)yx
+
(yz)xTT(yzx)
(yz,
x)
+
(x,
yz)
(x,
(x,
y)z
(zy)x
(z)yx
+TTT(yz)x
(yzx)++(yz,
(yz,
(x,
yz)
y)z
TT(zy)x
TT(z)yx
(z,
y)x(z,
(z,
(z,y)x
(z,x)y
x)y
=
TT(zy)
TT
(z)y
(z,
y)]x
=
[T
(yz)TT(y)z
(y,z)z)
z)++
+
T(zy)
(zy)
T(z)y
(z)y
(z,
(z,
y)]x
=[T
[T(yz)
(y)z(y,
(y,
y)]x
x)y]
[(xy,
z)z)
(x,
yz)
++
(x,
y)z]
[(zx,
y)(z,
+(z,
(z,
x)y]
[(xy,
z)
(x,
(x,
yz)
+(x,
(x,
y)z]
[(zx,
[(zx,y)
(z,xy)
xy)++
(z,
x)y]
[(xy,
yz)
y)z]
=
=
0.
=0.
0.
(4):
z)z)
==
ψ(x,
y)z
++
ψ(y,
z)x
ψ(z,
x)yx)y
(4):
By using
using(3),
getψ(xy,
ψ(xy,
z)
=
ψ(x,
y)z
+ψ(y,
ψ(y,
z)x
ψ(z,
ψ(z,
x)y
(4): By
(3),we
weget
get
ψ(xy,
ψ(x,
y)z
z)x
ψ(xy,
z)y
++
ψ(z,
y)x
ψ(y,
x)z
ψ(xz,
y).y).
Apψ(xy,
z) and
andψ(xz,
ψ(xz,y)y)==
=ψ(x,
ψ(x,
z)y
+
ψ(z,
y)x
ψ(y,
ψ(y,
x)z
ψ(xz,
ψ(xz,
y).
Apψ(xy,z)
ψ(x,
z)y
ψ(z,
y)x
x)z
Applying
with
the
fact
that
ψ ψisψis
anti
symmetric,
plying
theprevious
previoustwo
twoequations
equations
with
the
fact
that
isanti
anti
symmetric,
plyingthe
two
equations
with
the
fact
that
symmetric,
we
z)z)
ψ(xz,
y),y),
i.e.i.e.
2(ψ(xy,
z) z)
+z)++
we
have
ψ(xy,z)z)++ψ(xz,
ψ(xz,y)y)
y)==
=ψ(xy,
ψ(xy,
z)
ψ(xz,
y),
i.e.
2(ψ(xy,
we have
have ψ(xy,
ψ(xz,
ψ(xy,
ψ(xz,
2(ψ(xy,
ψ(xz,
free
ring,
ψ(xy,
z)z)
+
ψ(xz,
y)y)
=y)=
0.=0.0.
ψ(xz,
y))==0.0. Since
Risis
isaaa2-torsion
2-torsion
free
ring,
ψ(xy,
z)+
+ψ(xz,
ψ(xz,
ψ(xz,y))
SinceRR
2-torsion
free
ring,
ψ(xy,
(5):
This
part
is
achieved
from
(1),
(2)
and
(4),
immediately.
(5):
This
part
is
achieved
from
(1),
(2)
and
(4),
immediately.
(5):
achieved from (1), (2) and (4), immediately.
(6):
and
(5).
(6):
This part
partisisobtained
obtainedfrom
from(1)
(1)
and
(5).
(6): This
obtained
from
(1)
and
(5).
(7):
By
using
(2)
and
(5)
our
aim
is
achieved.
(7):
(5) our
our aim
aim isis achieved.
achieved.
(7): By using (2) and (5)
(8):
[z,[z,
w]r)
++
ψ(x,
y[z,
w]r)
(8):
We have
haveψ(x,
y)[z,w]r
w]r==
=ψ(xy,
ψ(xy,
[z,
w]r)
+ψ(x,
ψ(x,
y[z,
w]r)
(8): We
ψ(x,y)[z,
y)[z,
w]r
ψ(xy,
w]r)
y[z,
w]r)
=
w]zr)
===
0ψ([yz,
w]r,
x)+ψ([y,
w]zr,
x) x)x)
=
ψ([z,
w]r,xy)+ψ(x,
xy)+ψ(x,[yz,
[yz,w]r[y,
w]r[y,
w]zr)
0ψ([yz,
w]r,
x)+ψ([y,
w]zr,
=ψ([z,
ψ([z,w]r,
[yz,
w]r[y,
w]zr)
0ψ([yz,
w]r,
x)+ψ([y,
w]zr,
=
=
(see(7)).
(7)).
=000 (see
The
byby
Zalar
[11].
Now,
wewe
provide
The
followinglemma
hasbeen
beenproved
proved
by
Zalar
[11].
Now,
we
provide
The following
lemmahas
has
been
proved
Zalar
[11].
Now,
provide
another
another
prooffor
anotherproof
forit.it.
Lemma
ring
and
a a∈a∈R
element.
Lemma
2.8.Let
LetRRbebe
beaaasemiprime
semiprime
ring
and
∈Rbe
Rbeabeafixed
afixed
fixed
element.
Lemma 2.8.
semiprime
ring
and
element.
A GENERAL CHARACTERIZATION OF ADDITIVE ...
125
Then a[x, y] = 0 for all x, y ∈ R if and only if there exists an ideal I of
R contained in Z(R) such that a ∈ I.
Proof. First, note that [xy, z] = [x, z]y + x[y, z] and [x, yz] = [x, y]z +
y[x, z] for all x, y, z ∈ R. Let a[x, y] = 0 for some fixed element a and for
all x, y of R. We have [a, x]y[a, x] = ([ay, x] a[y, x])[a, x] = [ay, x][a, x]
and further, [ay[a, x], x] = [ay, x][a, x] + ay[[a, x], x] for all x, y ∈ R. So,
[a, x]y[a, x] = [ay, x][a, x]
= [ay[a, x], x] ay[[a, x], x]
= [a([ya, x] [y, x]a), x] a([y[a, x], x] [y, x][a, x])
= [a[ya, x] a[y, x]a, x] a[y[a, x], x] + a[y, x][a, x]
= 0.
Since R is semiprime, [a, x] = 0 for all x ∈ R, i.e. a ∈ Z(R). Suppose
I = < a > denotes the ideal generated by a . Hence, we have I =
{ra + as + na + Σm
i=1 ri asi | r, s, ri , si ∈ R, n ∈ Z, i = 1, 2, ..., m}.
Therefore, [ra + as + na + Σm
i=1 ri asi , x] = 0 and thus, a ∈ I ⊆ Z(R).
Conversely, assume that a ∈ I ⊆ Z(R), where I is a bi-ideal of R. We
will show that a[x, y] = 0 for all x, y ∈ R. Note that (a[x, y])z(a[x, y]) =
([ax, y] [a, y]x)z([ax, y] [a, y]x) = [ax, y]z[ax, y] = 0. Since R is
semiprime, a[x, y] = 0 for all x, y ∈ R and our purpose is achieved.
We are now ready for the following main result.
Theorem 2.9. Let R be a 2-torsion free semiprime ring and T : R → R
be a Jordan left centralizer associated with a l-semi Hochschild 2-cocycle
: R × R → R. Then, T is a left centralizer associated with .
Proof. By hypothesis,
T (x2 ) = T (x)x + (x, x) f or all x ∈ R.
(1)
Replacing x by x + y in (1), we get T (x2 ) + T (xy + yx) + T (y 2 ) =
T (x)x + T (x)y + T (y)x + T (y)y + (x, x) + (x, y) + (y, x) + (y, y).
This equation together with (1) imply that
T (xy + yx) = T (x)y + T (y)x + (x, y) + (y, x) f or all x, y ∈ R. (2)
126
A. HOSEINI
We replace y by xy + yx in (2) to get
T (x(xy + yx) + (xy + yx)x) = T (x)xy + T (x)yx + T (x)yx + T (y)x2 +
(x, y)x + (y, x)x + (x, xy) + (x, yx) + (xy, x) + (yx, x).
(3)
But this can also be calculated in a different way.
T (x2 y + yx2 ) + 2T (xyx) = T (x)xy + (x, x)y + T (y)x2 + (x2 , y)
+ (y, x2 ) + 2T (xyx) = T (x)xy + (x, x)y
+ T (y)x2 + (x, xy) (x, x)y + (yx, x)
+ (y, x)x + 2T (xyx).
It means that
T (x2 y + yx2 ) + 2T (xyx) = T (x)xy + (x, x)y + T (y)x2 + (x, xy)
(x, x)y + (yx, x) + (y, x)x + 2T (xyx).
(4)
Comparing (3) and (4), it is obtained that 2T (x)yx+(x, y)x+(x, yx)+
(xy, x) = 2T (xyx). Hence, 2T (x)yx + (x, y)x + (x, yx) + (x, yx)
(x, y)x = 2T (xyx). It means that
T (xyx) = T (x)yx + (x, yx).
(5)
Putting x + z for x in (5), we have T (x)yx + T (x)yz + T (z)yx + T (z)yz +
(x, yx)+(x, yz)+(z, yx)+(z, yz) = T (xyx)+T (xyz+zyx)+T (zyz).
The last equation together with (5) imply that
T (xyz + zyx) = T (x)yz + T (z)yx + (x, yz) + (z, yx).
(6)
Put m = T (xyzyx + yxzxy). We compute m in two different methods.
Using (5), we have
m = T (x)yzyx + (x, yzyx) + T (y)xzxy + (y, xzxy)
(7)
127
A GENERAL CHARACTERIZATION OF ADDITIVE ...
and by applying (6), we have
m = T (xy)zyx + T (yx)zxy + (xy, zyx) + (yx, zxy).
(8)
So,
0 = m m = T (xy)zyx + T (yx)zxy + (xy, zyx) + (yx, zxy)
T (x)yzyx (x, yzyx) T (y)xzxy (y, xzxy)
= T (xy)zyx + T (yx)zxy + (xy, zyx) + (yx, zxy)
T (x)yzyx T (y)xzxy (xy, zyx) (x, y)zyx
(yx, zxy) (y, x)zxy.
Hence, (T (xy)T (x)y (x, y))zyx+(T (yx)T (y)x(y, x))zxy = 0.
By the last equation and introducing a bilinear map ψ(x, y) = T (xy)
T (x)y (x, y), it can be achieved that
ψ(x, y)zyx + ψ(y, x)zxy = 0.
(9)
It follows from (2) that ψ(x, y) = ψ(y, x). Using this fact and equality
(9), we obtain
ψ(x, y)z[x, y] = 0 f or all x, y, z ∈ R.
(10)
Replacing x by x + u in (10), we get
0 = ψ(x + u, y)z[x + u, y]
= (ψ(x, y) + ψ(u, y))z([x, y] + [u, y])
= ψ(x, y)z[x, y] + ψ(x, y)z[u, y] + ψ(u, y)z[x, y] + ψ(u, y)z[u, y]
= 0 + ψ(x, y)z[u, y] + ψ(u, y)z[x, y] + 0.
Therefore,
ψ(x, y)z[u, y] + ψ(u, y)z[x, y] = 0 f or all x, y, z, u ∈ R.
Using (10) and (11), we find
(ψ(x, y)z[u, y])ω(ψ(x, y)z[u, y]) = ψ(x, y)(z[u, y]ωψ(x, y)z)[u, y]
= ψ(u, y)z[u, y]ωψ(x, y)z[x, y]
= 0.
(11)
128
A. HOSEINI
Since R is semiprime, we obtain
ψ(x, y)z[u, y] = 0 f or all x, y, z, u ∈ R.
(12)
Replacing y by y + v in (12), we arrive at
ψ(x, y)z[u, y] + ψ(x, y)z[u, v] + ψ(x, v)z[u, y] + ψ(x, v)z[u, v] = 0,
this equation together with (12) imply that
ψ(x, y)z[u, v] = ψ(x, v)z[u, y] f or all x, y, z, u, v ∈ R.
(13)
From (13) and the fact that ψ(x, y)z[u, y] = 0, it can be concluded that
(ψ(x, y)z[u, v])ω(ψ(x, y)z[u, v]) = ψ(x, y)(z[u, v]ωψ(x, y)z)[u, v]
= ψ(x, v)z[u, v]ωψ(x, y)z[u, y]
= 0.
Now, we have
ψ(x, y)z[u, v]ωψ(x, y)z[u, v] = ψ(x, y)(z[u, v]ωψ(x, y)z)[u, v]
= ψ(x, v)(z[u, v]ωψ(x, y)z)[u, y]
(see 13)
= 0.
(see 12)
= ψ(x, v)z[u, v]ω(ψ(x, y)z[u, y])
Reusing the fact that R is semiprime, it seen that
ψ(x, y)z[u, v] = 0 f or all x, y, z, u, v ∈ R.
(14)
Hence,
ψ(x, y)[u, v]zψ(x, y)[u, v] = ψ(x, y)([u, v]zψ(x, y))[u, v] = 0,
(see 14)
and it follows from semiprimeness of R that
ψ(x, y)[u, v] = 0 f or all x, y, u, v ∈ R.
(15)
Now, let x and y be two fixed elements of R and for convenience write
ψ instead of ψ(x, y). Using Lemma 1 we get the existence of an ideal I
AAGENERAL
OFOF
ADDITIVE
... ...
GENERALCHARACTERIZATION
CHARACTERIZATION
ADDITIVE
129129
such
ψyψy
∈ Z(R)
for for
all all
y ∈yR.
such that
that ψψ ∈∈II⊆⊆Z(R).
Z(R).InInparticular,
particular,yψ,
yψ,
∈ Z(R)
∈ R.
This
gives
us
This gives us
2
2 2
2 2
2 2
yψyψ
.x .x
==
y.ψy.ψ
x x
x.ψ
x.ψy2 y==ψψy.x
y.x==
(16)(16)
2
2 2
and
of of
this
equality
willwill
be be
and so,
so, 4T
4T(x.ψ
(x.ψ 2y)y)==4T4T(y.ψ
(y.ψx).
x). Both
Bothsides
sides
this
equality
computed
using
(2)
and
(16).
Indeed,
we
have
computed using (2) and (16). Indeed, we have
2 2
2 2
4T4T(x.ψ
4T4T
(y.ψ
(x.ψy)
y)==
(y.ψx).
x).
2
2 2
2
2 2
⇒
2T2T
(yψ
xy).
⇒2T
2T(xψ
(xψy2 y++ψ ψyx)
yx)= =
(yψx2+
xψ
+ψ
xy).
2
2 2
2 2
2 2
⇒2T
++
2T2T
(ψ(ψ
y)x
+ 2(ψ
y, x)
⇒2T(x)ψ
(x)ψy2 y++2(x,
2(x,ψψy)
y)
y)x
+ 2(ψ
y, x)
22
2 2
2 2
2 2
ψ ψx)x)
++
2T2T
(ψ(ψx)yx)y
+ 2(ψ
x, y).
==2T
2T(y)ψ
(y)ψxx++2(y,
2(y,
+ 2(ψ
x, y).
2
22
2
2 2
2 2
⇒2T
T (ψ
yψyψ
)x )x
+ 2(ψ
y, x)
⇒2T(x)ψ
(x)ψ 2yy++2(x,
2(x,ψψy)
y)++
T (ψy2+
y+
+ 2(ψ
y, x)
22
2 2
2 2
2 2
2 2
++
T (ψ
xψxψ
)y )y
+ 2(ψ
x, y).
==2T
2T(y)ψ
(y)ψxx++2(y,
2(y,ψψx)x)
T (ψx +
x+
+ 2(ψ
x, y).
2
22
2 2
2 2
2 2
⇒2T
(ψ(ψ
)yx
++
(ψ
+ T+(y)ψ
x +x(y,
ψ 2 )x
⇒2T(x)ψ
(x)ψ 2yy++2(x,
2(x,ψψy)
y)++T T
)yx
(ψ, y)x
, y)x
T (y)ψ
+ (y,
ψ 2 )x
2
2
2 + T (ψ)ψxy + (ψ, ψ)xy
ψ 2ψx)
+
+2(ψ
2(ψ 2y,y,x)
x)==2T
2T(y)ψ
(y)ψx2 x++2(y,
2(y,
x) + T (ψ)ψxy + (ψ, ψ)xy
2
2
2 2
(x,
ψ 2ψ)y2 )y
+ 2(ψ
x, y).
++(ψ
(ψ,2x)y
, x)y++TT(x)ψ
(x)ψy2 +
y+
(x,
+ 2(ψ
x, y).
2
22
2 2
⇒T
(ψ)ψyx
++
(ψ,
ψ)yx
+ (ψ
, y)x
+ (y,
ψ 2 )x
⇒T(x)ψ
(x)ψ 2yy++2(x,
2(x,ψψy)
y)++T T
(ψ)ψyx
(ψ,
ψ)yx
+ (ψ
, y)x
+ (y,
ψ 2 )x
2
2
2(y,
ψ 2ψx)2 x)
++
T (ψ)ψxy
+ (ψ,
ψ)xy
+
+2(ψ
2(ψ 2y,y,x)
x)==TT(y)ψ
(y)ψx2 x++
2(y,
T (ψ)ψxy
+ (ψ,
ψ)xy
2
2 2
++(ψ
ψ 2ψ)y2 )y
++
2(ψ
x, y).
(ψ,2x)y
, x)y++(x,
(x,
2(ψ
x, y).
Since
Since ψyx
ψyx==xψy
xψy==ψxy,
ψxy,we
weobtain
obtain
2
22
2 2
2 2
++
(ψ
+ (y,
ψ 2 )x
y, x)y,=
TT(x)ψ
(x)ψ 2yy++2(x,
2(x,ψψy)
y)++(ψ,
(ψ,ψ)yx
ψ)yx
(ψ, y)x
, y)x
+ (y,
ψ 2+
)x2(ψ
+ 2(ψ
x) =
2
22
2 2
2 2
++
(ψ
+ (x,
ψ 2 )y
x, y).
TT(y)ψ
(y)ψ 2xx++2(y,
2(y,ψψx)
x)++(ψ,
(ψ,ψ)xy
ψ)xy
(ψ, x)y
, x)y
+ (x,
ψ 2+
)y2(ψ
+ 2(ψ
x, y).
(17)(17)
130
130
130
130
130
130
A. A.
HOSEINI
HOSEINI
A.A.
HOSEINI
A.
HOSEINI
HOSEINI
A. HOSEINI
Rearranging
Rearranging(17)
(17)wewegetget
Rearranging
(17)
we
get
Rearranging
Rearranging(17)
(17)we
weget
get
Rearranging (17) 2we
get
2 2
2
2
2 + (y, ψ 2 )x+2
TT(y)xψ
y+
2(x,
ψ 2ψ
y)22 y)
+ (ψ,
ψ)yx
+ (ψ
, y)x
(y)xψ222 ==TT(x)ψ
(x)ψ
y+
2(x,
+ (ψ,
ψ)yx
+ 2(ψ
, y)x + (y,
2
2 ψ2 )x+
222y + 2(x, ψ22 y)
2
=
TT(x)ψ
(x)ψ
+(ψ,
(ψ,
ψ)yx
+(ψ
(ψ
, y)x
(y,
)x+
TT(y)xψ
(y)xψ
=
T
y
+
2(x,
ψ
y)
+
ψ)yx
+
,
y)x
++(y,
ψ 2ψ)x+
T
(y)xψ
=
(x)ψ
y
+
2(x,
ψ
y)
+
(ψ,
ψ)yx
+
(ψ
,2y)x
+ (y,
ψ )x+
2
2
2
2
T (y)xψ = T (x)ψ y + 2(x, ψ y) + (ψ, ψ)yx + (ψ , y)x + (y, ψ )x+
2
2 (ψ, ψ)xy (ψ 2 , x)y
2 (x, ψ 2 )y
2
ψ22ψ
x)
2(ψ
x, y).
2(ψ
2(ψ22222y,y,x)x)2(y,
2(y,
x) (ψ, ψ)xy (ψ
, x)y (x,
ψ 2 )y
2(ψ
x, y).
2 2 , 2x)y (x, ψ
2 2 )y2 2(ψ
2 2 x,2y).
2 (ψ, ψ)xy (ψ
y,
x)
2(y,
2(y,
ψψ2x)
x)
2(ψ
y,
x)
ψ
(ψ,
ψ)xy
(ψ
,
x)y
(x,
ψ
)y
2(ψ
x,
y).
2(ψ
2(ψ
y,
x)
2(y,
x)
(ψ,
ψ)xy
(ψ
,
x)y
(x,
ψ
)y
2(ψ
x, y).
(18)
2
2
2
2
2
(18)
2(ψ y, x) 2(y, ψ x) (ψ, ψ)xy (ψ , x)y (x, ψ )y 2(ψ x, y). (18)
(18)
(18)
(18)
On
also
have
Onthe
theother
otherhand,
hand,wewe
also
have
On
the
other
hand,
we
also
have
On
have
On the
theother
otherhand,
hand,we
wealso
also
have
On the other hand, we also have
2 2
4T4T
(xyψ
) =) 4T
(xψ.yψ)
(xyψ
= 4T
(xψ.yψ)
22 )2 = 4T (xψ.yψ)
4T
(xyψ
)
=
4T
(xψ.yψ)
4T
(xyψ
4T
(xyψ
)
=
4T
(xψ.yψ)
4T (xyψ 2 ) = 4T (xψ.yψ)
2 2 2 2
++
ψψ
xy)
= 2T
+ yψxψ)
⇒⇒2T2T
(xyψ
(xyψ
xy)
= (xψyψ
2T (xψyψ
+ yψxψ)
22 2+ ψ22 xy)
2
(xψyψ
yψxψ)
⇒
2T
(xyψ
++
ψ ψxy)
==2T
(xψyψ
++yψxψ)
⇒
2T
(xyψ
⇒
2T
(xyψ
xy)
=2T2T
(xψyψ
+
yψxψ)
2
2
⇒ 2T (xyψ + ψ xy) = 2T (xψyψ + yψxψ)
2 2
2 2
2 2(ψ 2 , xy)
2
⇒2T
++
2T2T
(ψ(ψ
)xy
+ 2(xy,
ψ222)ψ+
⇒2T(xy)ψ
(xy)ψ
)xy
+ 2(xy,
) + 2(ψ
, xy)
222+ 2T (ψ22 )xy
2 2 , xy)
2
2
+2(xy,
2(xy,
ψ)ψ)+2+
2(ψ
⇒2T
(xy)ψ
+
2T
(ψ
)xy
+
ψ
2(ψ
,
xy)
⇒2T
(xy)ψ
⇒2T
(xy)ψ
+
2T
(ψ
)xy
+
2(xy,
)
+
2(ψ
, xy)
2 = 2T (ψx)ψy
2
2
2
2T2T
(ψy)ψx
2(ψx,
ψy)ψy)
+ 2(ψy,
ψx) ψx)
= 2T
(ψx)ψy
+
(ψy)ψx
+ 2(ψx,
+ 2(ψy,
(ψ
)xy ++2(xy,
ψ ) +++2(ψ
, xy)
⇒2T (xy)ψ +
==2T
2T
(ψx)ψy
2T
(ψy)ψx
ψy)
2(ψy,
ψx)
=
2T
2T
(ψy)ψx
+ 2(ψx,
ψy)
++2(ψy,
ψx)
2T(ψx)ψy
(ψx)ψy+++
2T
(ψy)ψx
+2(ψx,
2(ψx,
ψy)
+
2(ψy,
ψx)
= 2T (ψx)ψy + 2T (ψy)ψx + 2(ψx, ψy) + 2(ψy, ψx)
22
2
2
⇒2T
++2T2T
(ψ)ψxy
+ 2(ψ,
ψ)xy
+ 2(xy,
ψ22 2) ψ
+22(ψ
, xy)
⇒2T(xy)ψ
(xy)ψ
(ψ)ψxy
+ 2(ψ,
ψ)xy
+ 2(xy,
) + 2(ψ
, xy)
2
222+ 2T (ψ)ψxy + 2(ψ, ψ)xy + 2(xy, ψ
2 2 , xy)
⇒2T
(xy)ψ
) 2+2(ψ
2(ψ
⇒2T
(xy)ψ
⇒2T
(xy)ψ
+
2T
(ψ)ψxy
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
2(ψ
, xy)
+
2T
(ψ)ψxy
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
,
xy)
2
2
2
==2T
++
2T2T
(ψy)ψx
2(ψx,
+ 2(ψy,
2T(ψx)ψy
(ψx)ψy
(ψy)ψx
+ 2(ψx,
+ 2(ψy,
ψx)
⇒2T (xy)ψ + 2T
(ψ)ψxy
+ 2(ψ,
ψ)xy
++
2(xy,
ψ ψy)
)ψy)
+ψy)
2(ψ
, xy)ψx)
=
2T
(ψx)ψy
2T
(ψy)ψx
2(ψy,
ψx)
=2T
2T(ψx)ψy
(ψx)ψy+++
2T
(ψy)ψx
+2(ψx,
2(ψx,
ψy)
+
2(ψy,
ψx)
=
2T
(ψy)ψx
++2(ψx,
ψy)
++2(ψy,
ψx)
= 2T (ψx)ψy + 2T (ψy)ψx + 2(ψx, ψy) + 2(ψy, ψx)
22
2
2
++2T2T
(ψ)ψxy
+ 2(ψ,
ψ)xy
+ 2(xy,
ψ22 2) ψ
+22(ψ
, xy)
⇒2T
⇒2T(xy)ψ
(xy)ψ
(ψ)ψxy
+ 2(ψ,
ψ)xy
+ 2(xy,
) + 2(ψ
, xy)
222+ 2T (ψ)ψxy + 2(ψ, ψ)xy + 2(xy, ψ
2 2 , xy)
2
) 2+2(ψ
2(ψ
⇒2T
(xy)ψ
+
2T
(ψ)ψxy
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
,
xy)
⇒2T
(xy)ψ
+
2T
(ψ)ψxy
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
2(ψ
,
xy)
⇒2T
(xy)ψ
2
==2T
TT(ψx
++
Tψ)xy
(yψ
++ψy)ψx
+ψ2(ψx,
ψy)2 ,ψy)
+
2(ψy,
ψx) ψx)
(ψx++xψ)ψy
xψ)ψy
T (yψ
+2(xy,
ψy)ψx
+
2(ψx,
+ 2(ψy,
(ψ)ψxy
+
2(ψ,
)
+
2(ψ
xy)
⇒2T (xy)ψ 2 +
=
(ψx
++xψ)ψy
xψ)ψy
2(ψx,
ψy)
2(ψy,
ψx)ψx)
=
TT(yψ
++ψy)ψx
++2(ψx,
ψy)
++2(ψy,
ψx)
=TTT(ψx
(ψx+
xψ)ψy+++
T(yψ
(yψ
+ψy)ψx
ψy)ψx
+
2(ψx,
ψy)
+
2(ψy,
= T (ψx + xψ)ψy + T (yψ + ψy)ψx + 2(ψx, ψy) + 2(ψy, ψx)
22
2
2
⇒2T
++2T2T
(ψ)ψxy
+ 2(ψ,
ψ)xy
+ 2(xy,
ψ22 2) ψ
+22(ψ
, xy)
⇒2T(xy)ψ
(xy)ψ
(ψ)ψxy
+ 2(ψ,
ψ)xy
+ 2(xy,
) + 2(ψ
, xy)
2
222+ 2T (ψ)ψxy + 2(ψ, ψ)xy + 2(xy, ψ
2 2 , xy)
) 2+2(ψ
2(ψ
⇒2T
(xy)ψ
⇒2T
(xy)ψ
⇒2T
(xy)ψ
+
2T
(ψ)ψxy
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
2(ψ
,
xy)
+
2T
(ψ)ψxy
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
,
xy)
2
2
2 + (x, ψ)ψy + (ψ, x)ψy +
2 T (ψ)ψxy2 + T (y)ψ x 2
y
==2T
TT(x)ψ
(x)ψ
y
+
(x,
ψ)ψy
+
(ψ,
x)ψy
+
T
(ψ)ψxy
+
T
(y)ψ
x
⇒2T (xy)ψ 2 +
(ψ)ψxy
+
2(ψ,
ψ)xy
2(xy,
ψ
)
2(ψ
,
xy)
2
2
2 x2
=
(x)ψ
++(x,
(x,
ψ)ψy
(ψ,
x)ψy
T(ψ)ψxy
(ψ)ψxy
T
(y)ψ
ψ)ψy
+++
(ψ,
x)ψy
++T+
++T +
(y)ψ
x x
=TTT(x)ψ
(x)ψ2 y2yy+
(x,
ψ)ψy
(ψ,
x)ψy
T (ψ)ψxy
T (y)ψ
=
2
++(y,
ψ)ψx
+T
+ (ψ,
y)ψx
(y,
ψ)ψx
+(ψ)ψxy
T (ψ)ψxy
++(ψ,
y)ψx
+
(ψ,
x)ψy
+
T
(ψ)ψxy
T
(y)ψ
x
= T (x)ψ 2 y + (x, ψ)ψy
+
(y,
ψ)ψx
+
T
(ψ)ψxy
+
(ψ,
y)ψx
++(y,
ψ)ψx
++
T (ψ)ψxy
+ (ψ,
y)ψx
(y,
ψ)ψx
T (ψ)ψxy
+ (ψ,
y)ψx
+(ψ)ψxy
2(ψx,
ψy)
+ 2(ψy,
ψx)ψx)
+
2(ψx,
ψy)
+
2(ψy,
+ (y, ψ)ψx + T
+
(ψ,
y)ψx
2(ψx,
ψy)
ψx)
+++
2(ψx,
ψy)
++2(ψy,
ψx)
2(ψx,
ψy)
+2(ψy,
2(ψy,
ψx)
+ 2(ψx, ψy) + 2(ψy, ψx)
22
2 2(ψ 2 , xy)
2 = T (x)ψ 2 y +2(x, ψ)ψy
⇒2T
++2(ψ,
ψ)xy
++
2(xy,
ψ222)ψ2+
⇒2T(xy)ψ
(xy)ψ
2(ψ,
ψ)xy
2(xy,
) + 2(ψ
, xy) = T (x)ψ
y + (x, ψ)ψy
222+ 2(ψ, ψ)xy + 2(xy, ψ
2 2, 2
2 2 y 2+ (x, ψ)ψy
)++
2(ψ
xy)
T
(x)ψ
⇒2T
(xy)ψ
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
2(ψ
,
xy)
==T=
(x)ψ
y + (x,
ψ)ψyψ)ψy
⇒2T
(xy)ψ
⇒2T
(xy)ψ
+
2(ψ,
ψ)xy
+
2(xy,
ψ
)
+
2(ψ
, xy)
T (x)ψ
y + (x,
2
2
2
2
⇒2T (xy)ψ + 2(ψ, ψ)xy + 2(xy, ψ ) + 2(ψ , xy) = T (x)ψ y + (x, ψ)ψy
2 2
++
(y,
ψ)ψx
+ (ψ,
y)ψx
+ 2(ψx,
ψy) ψy)
+ 2(ψy,
ψx) ψx)
+(ψ,
(y)xψ
+(ψ,x)ψy
x)ψy++T T
(y)xψ
(y,
ψ)ψx
+ (ψ,
y)ψx
+ 2(ψx,
+ 2(ψy,
22 2+ (y, ψ)ψx + (ψ, y)ψx + 2(ψx, ψy) + 2(ψy, ψx)
+(ψ,
x)ψy
++TTT(y)xψ
(y)xψ
++
(y,
ψ)ψx
+ (ψ,
y)ψx
+ 2(ψx,
ψy) ψy)
+ 2(ψy,
ψx)
+(ψ,
+(ψ,x)ψy
x)ψy+
(y)xψ
(y,
ψ)ψx
+ (ψ,
y)ψx
+ 2(ψx,
+
2(ψy,
ψx)
(19)
2
(19)
+(ψ, x)ψy + T (y)xψ + (y, ψ)ψx + (ψ, y)ψx + 2(ψx, ψy) + 2(ψy, ψx) (19)
(19)
(19)
(19)
A GENERAL CHARACTERIZATION OF ADDITIVE ...
AA
GENERAL
GENERAL
GENERAL
CHARACTERIZATION
CHARACTERIZATION
CHARACTERIZATION
OF
OF
OF
ADDITIVE
ADDITIVE
ADDITIVE
.........
131
131
131
131
Using (18) and (19), we have
Using
Using
(18)
(18)
and
and
and
(19),
(19),
(19),
wewe
we
have
have
have
2T (xy)ψ 22 + 2(ψ, ψ)xy + 2(xy, ψ 22 ) + 2(ψ 22 , xy) = T (x)ψ 22 y + (x, ψ)ψy
2 22
2 22
2 22
22 2
2(ψ,
ψ)xy
2(xy,
))++
2(ψ
, xy)
(x,
ψ)ψy
2T2T
(xy)ψ
(xy)ψ
++
+
2(ψ,
2(ψ,
ψ)xy
ψ)xy
++
+
2(xy,
2(xy,
ψψ
ψ
)+
2(ψ
2(ψ
, xy)
, xy)
===
TT(x)ψ
T(x)ψ
(x)ψ
yy+
y++
(x,
(x,
ψ)ψy
ψ)ψy
+ (ψ, x)ψy + (y, ψ)ψx + (ψ, y)ψx + 2(ψx, ψy) + 2(ψy, ψx)
2+
2 ψy)
2ψx)
(ψ,
x)ψy
(y,
ψ)ψx
(ψ,
y)ψx
2(ψx,
2(ψy,
++
(ψ,
x)ψy
x)ψy
+2 +
+
(y,
(y,
ψ)ψx
ψ)ψx
+
+
(ψ,
(ψ,
y)ψx
y)ψx
+++
2(ψx,
2(ψx,
ψy)
ψy)
+++
2(ψy,
2(ψy,
ψx)
ψx)
+ T (x)ψ 2 y + 2(x, ψ 2 y) + (ψ, ψ)yx + (ψ 2 , y)x + (y, ψ 2 )x
2 22
2 22
222
22 2
2
T
++
T (x)ψ
(x)ψ
y+
+
2(x,
2(x,
y)y)
y)
+ψ+22+
(ψ,
(ψ,
ψ)yx
+
(ψ
(ψ
y)x
, y)x
+
(y,
(y,
ψψψ
)x
y22yy,
+
2(x,
ψψ
(ψ,
ψ)yx
++
(ψ
, y)x
(y,
)x)x
2+
x)
ψ
2(y,
x)
ψ)yx
(ψ,
ψ)xy
,(ψ
,+
x)y
+(x)ψ
2(ψ
2 22
2 22
22 2
2 x)
2ψ)xy
2(y,
(ψ,
(ψ
x)y
++
2(ψ
2(ψ
y,y,
x)
y,x)
x)
2(y,
2(y,
ψ ψψ
x)
x)
(ψ,
(ψ,
ψ)xy
ψ)xy
(ψ
(ψ
, ,x)y
, x)y
2(ψ
(x, ψ 2 )y 2(ψ 2 x, y)
22
2 22
)y
2(ψ
(x,
(x,
(x,
ψ 2ψψ
)y
)y
2(ψ
2(ψ
x,x,x,
y)y)y)
Hence,
Hence,
Hence, 2
2T (xy)ψ 2 2T (x)yψ 22 = (x, ψ)ψy + (ψ, x)ψy + (y, ψ)ψx + (ψ, y)ψx
2 22
2 22
2
(xy)ψ
(x)yψ
=
(x,
ψ)ψy
x)ψy
(y,
ψ)ψx
(ψ,
y)ψx
2T2T
(xy)ψ
2T2T
2T
(x)yψ
(x)yψ
=+
=
(x,
(x,
ψ)ψy
ψ)ψy
+++
(ψ,
(ψ,
x)ψy
x)ψy
+++
(y,
(y,
ψ)ψx
ψ)ψx
(ψ,
(ψ,
y)ψx
y)ψx
2+
y)+
2(ψx,
ψy)
+(ψ,
2(ψy,
ψx)
+
2(x,
ψ+
22 2
2 ψx)
2 ψψψ
2(ψx,
ψy)
2(ψy,
ψx)
2(x,
++
+
2(ψx,
2(ψx,
ψy)
ψy)
+++
2(ψy,
2(ψy,
ψx)
+++
2(x,
2(x,
y)
y)y) 2
+ (ψ, ψ)yx + (ψ 2 , y)x + (y, ψ 2 )x + 2(ψ 2 y, x)
2 22
22 2
22 2
2
, ψ)xy
2(ψ
++
+
(ψ,
(ψ,
ψ)yx
ψ)yx
++
+
(ψ
(ψ
, y)x
,y)x
y)x
+++
(y,
(y,
ψψ22ψ
)x
)x
++
+
2(ψ
2(ψ
y,y,
y,
(ψ,
ψ)yx
(ψ
2x)
2(y,
ψ 22 x)
(ψ,
(y,
(ψ
, )x
x)y
(x,
ψx)
)yx)
2 22
22 2
22 2
2
(ψ,
ψ)xy
x)y
(x,
)y)y
2(y,
2(y,
2(y,
ψ22ψx,
ψ
x)x)
x)
(ψ,
(ψ,
ψ)xy
ψ)xy
(ψ
(ψ
, ,x)y
, x)y
(x,
(x,
ψψψ
)y
2 )
y)
2(ψ,
ψ)xy
(ψ
2(xy,
ψ
2(ψ
2 22
22 2
2x,
2(ψ
2(ψ
2(ψ
x,
x,
y)y)
y)
2(ψ,
2(ψ,
ψ)xy
ψ)xy
2(xy,
2(xy,
ψψψ
)) )
2(ψ,
ψ)xy
2(xy,
2(ψ 2 , xy)
()
2 22
,,xy)
()
2(ψ
2(ψ
, xy)
xy) ()
()
2(ψ
Our next task is to prove the equation bellow:
next
task
prove
the
equation
bellow:
Our
Our
next
task
task
is is
is
toto
to
prove
prove
the
the
equation
equation
bellow:
bellow:
2(T (xy) T (x)y (x, y))ψ 22 = 0
2 22
2(T
(xy)
(x,
y))ψ
2(T
2(T
(xy)
(xy)
TT
(x)y
T(x)y
(x)y
(x,
(x,
y))ψ
y))ψ
===
000
In order to prove the previous equation we need the following relations:
order
prove
the
previous
equation
we
need
the
following
relations:
In In
order
toto
prove
prove
the
the
previous
previous
equation
equation
we
we
need
need
the
the
following
following
relations:
relations:
(ψ, x)ψy = (ψ, xψ)y (ψx, ψ)y = (ψ, ψx)y (ψx, ψ)y
2
(ψ,
x)ψy
(ψ,
(ψx,
ψ)y
==
(ψ,
ψx)y
(ψx,
ψ)y
(ψ,
x)ψy
x)ψy
==
=
(ψ,
(ψ,
xψ)y
(ψx,
(ψx,
ψ)y
ψ)y
=
(ψ,
(ψ,
ψx)y
ψx)y
(ψx,
(ψx,
ψ)y
ψ)y
2 xψ)y
=
(ψ
,xψ)y
x)y+
(ψ,
ψ)xy
(ψx,
ψ)y
(i)
2 22
,,x)y
(ψ,
ψ)xy
(ψx,
ψ)y
(i)(i)
(ψ
==
=
(ψ
(ψ
, x)y
x)y
+++
(ψ,
(ψ,
ψ)xy
ψ)xy
(ψx,
(ψx,
ψ)y
ψ)y
(i)
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
2(ψx, ψy) = (ψx + xψ, ψy) = (ψx, ψy) + (xψ, ψy)
2 +
ψy)
(ψx
xψ,
ψy)
(ψx,
ψy)
(xψ,
ψy)
2(ψx,
2(ψx,
ψy)
==
=
(ψx
(ψx
+
+
xψ,
xψ,
ψy)
ψy)
===
(ψx,
(ψx,
ψy)
ψy)
+++
(xψ,
(xψ,
ψy)
ψy)
= (ψ 2 x, y) + (ψx, ψ)y + (xψ, ψy)
2 22
2ψ)y
(ψ
(ψx,
ψ)y
(xψ,
ψy)
==
=
(ψ
(ψ
x,2x,
x,
y)y)
y)
++
+
(ψx,
(ψx,
ψ)y
+++
(xψ,
(xψ,
ψy)
ψy)
(ii)
= (ψ 2 x, y) + (x, ψ 2 )y (x, ψ)ψy + (xψ, ψy)
2 22
2 22
(x,
(x,
ψ)ψy
(xψ,
ψy)
(ii)
(ψ
==
=
(ψ
(ψ
x,x,
x,
y)y)
y)
++
+
(x,
(x,
ψ ψψ
)y)y
)y
(x,
(x,
ψ)ψy
ψ)ψy
+++
(xψ,
(xψ,
ψy)
ψy)
(ii)
(ii)
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
(ψ, y)ψx = (ψ, yψx) (ψy, ψx) = (ψ, ψxy) (ψy, ψx)
2
y)ψx
(ψ,
(ψy,
ψx)
==
(ψ,
ψxy)
(ψy,
ψx)
(ψ,
(ψ,
y)ψx
==
=
(ψ,
(ψ,
yψx)
(ψy,
(ψy,
ψx)
ψx)
=
(ψ,
(ψ,
ψxy)
ψxy)
(ψy,
(ψy,
ψx)
ψx)
2 yψx)
,yψx)
xy)+
(ψ,
ψ)xy
(ψy,
ψx)
=
(ψ
2 22
(ψ
,,xy)
(ψ,
ψ)xy
(ψy,
ψx)
==
=
(ψ
(ψ
, xy)
xy)
++
+
(ψ,
(ψ,
ψ)xy
ψ)xy
(ψy,
(ψy,
ψx)
ψx)
(iii)
(iii)
(iii)
(iii)
132
132
132
132
A. HOSEINI
A.
A. HOSEINI
HOSEINI
A.
A.HOSEINI
HOSEINI
2(ψy, ψx)
ψx) =
= 2(yψ,
2(yψ, ψx) =
= 2(y,ψψ222x)
x) 2(y,
2(y, ψ)ψx
(iv)
2(ψy,
(iv)
2(yψ, ψx)
ψx) = 2(y,
2(y, ψ22x)
2(y,ψ)ψx
ψ)ψx
(iv)
2(ψy,
(iv)
2(ψy,ψx)
ψx) =
= 2(yψ,
2(yψ,ψx)
ψx)==2(y,
2(y,ψψ x)
x)2(y,
2(y,ψ)ψx
ψ)ψx
(iv)
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
2
(ψ, ψ)yx
ψ)yx =
= (ψ,
(ψ, ψyx)
(ψ222,,yx)
yx) = (ψ,
(ψ, ψxy)
(ψ222,,yx)
yx)
(ψ,
(ψ, ψyx)
ψyx) (ψ
(ψ22, yx) =
= (ψ,ψxy)
ψxy) (ψ
(ψ2 2, yx)
2ψyx)
2ψxy)
,
yx)
=
(ψ,
(ψ
,
yx)
(ψ,
(ψ,
(ψ
(ψ,ψ)yx
ψ)yx =
=
(ψ,
ψyx)
(ψ
,
yx)
=
(ψ,
ψxy)
(ψ
,
yx)
xy) + (ψ,
(ψ, ψ)xy
(ψ22,,yx)
yx)
= (ψ
(ψ22,, xy)
=
, xy) +
+ (ψ,ψ)xy
ψ)xy (ψ
(ψ2 2, yx)
222
=
= (ψ
(ψ22,,,xy)
xy)+
+(ψ,
(ψ,ψ)xy
ψ)xy
(ψ
(ψ222,y,
, yx)
yx)
xy)
+
(ψ,
ψ)xy
(ψ
x)
(ψ
(ψ222,,y)x
y)x
(v)
=
(ψ
,
xy)
+
(ψ,
ψ)xy
(ψ
y,
x)
(v)
=
(ψ
,
xy)
+
(ψ,
ψ)xy
(ψ
y,
x)
(ψ
, y)x
(v)
22
22
22
(v)
= (ψ
(ψ ,,xy)
xy)++(ψ,
(ψ,ψ)xy
ψ)xy(ψ
(ψy,y,x)x)(ψ
(ψ, y)x
, y)x
(v)
=
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
2
2
(ψ22y,
y, x)
x) =
= (yψ
(yψ222,, x)
x) =
= (y,
(y,ψψ222x)
x)
(y,
(y,ψψ222)x
)x
(vi)
(ψ
(vi)
(yψ
,
x)
=
(y,
ψ
x)
(y,
ψ2 2)x
(vi)
22
22
22
(vi)
(ψ
(ψ y,
y,x)
x) =
= (yψ
(yψ ,,x)
x)==(y,
(y,ψψ x)
x)(y,
(y,ψψ)x)x
(vi)
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
2
2
2
2
2(xy, ψ
ψ222)) =
= 2(x,
2(x, yψ
yψ222)) +
+ 2(x,
2(x,y)ψ
y)ψ222
(vii)
(vii)
2(xy,
2(x,
yψ
)
+
2(x,
y)ψ
(vii)
22
22
22
(vii)
2(xy,
2(xy,ψψ )) =
= 2(x,
2(x,yψ
yψ ))++2(x,
2(x,y)ψ
y)ψ
(vii)
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
x, y)
y) =
= (ψx,
(ψx, ψy) +
+ (ψx,ψ)y
ψ)y
(viii)
(ψ22x,
(ψ
(viii)
(ψx, ψy)
ψy) + (ψx,
(ψx, ψ)y
(viii)
22
(viii)
(ψ
(ψ x,
x,y)
y) =
= (ψx,
(ψx,ψy)
ψy)++(ψx,
(ψx,ψ)y
ψ)y
(viii)
———————————————————————————————–
———————————————————————————————–
———————————————————————————————–
By
using the
the above
above eight relations,
relations, the equation
equation () turns
turns into
———————————————————————————————–
———————————————————————————————–
By
using
above eight
eight relations, the
the equation ()
() turns into
into
By
()
By using
using the
the above
above eight
eightrelations,
relations,the
theequation
equation
()turns
turnsinto
into
2
2(T(xy)
(xy)
TT(x)y
(x)y
(x,
(x,y))ψ
y))ψ22 =
=0.
0.
2(T
2(T (xy) T (x)y (x, y))ψ2 2 = 0.
2(T
2(T(xy)
(xy)TT(x)y
(x)y(x,
(x,y))ψ
y))ψ ==0.0.
Since R
R is
is aa 2-torsion
2-torsion free
free ring,
ring, the
the above
above equation
equation reduces
reduces to(T
(T (xy)
Since
2-torsion
free
ring,
the
above
equation
reducesto
to (T2(xy)
(xy)
2 = free
3 =reduces
2 =
0,
and
this
means
that
ψ
0.
Hence,
ψ
Rψ
T
(x)y
(x,
y))ψ
Since
R
is
a
2-torsion
ring,
the
above
equation
to
(T
(xy)
Since
R
is
a
2-torsion
free
ring,
the
above
equation
reduces
to
(xy)
22 = 0, and this means that ψ 33 = 0. Hence, ψ 2(T
22
2
T (x)y (x, y))ψ
Rψ
=
y))ψ22 = 0, and this means that ψ3 3 = 0. Hence, ψ2 2Rψ
=
2
2
4
3
2
ψ
R=
=
ψψ
Ry))ψ
= {0}.
{0}.
semiprimeness
R
that
ψRψ
==0.=
T
(x,
==Thus,
0,0, and
this
means
ψψRψ
T(x)y
ψψ
(x,
y))ψ
andthe
this
meansthat
thatψof
ψof=
=0.implies
0.Hence,
Hence,
4(x)y
33R
ψ
R
=
{0}. Thus,
Thus, the
the semiprimeness
semiprimeness
ofR
Rimplies
impliesthat
thatψψ2222==0.0.
4
3
4
3
2
ψ
=
ofofRRimplies
that
ψψR=is=
ψR
R=
= ψψ
ψψ R
Rwe
= {0}.
{0}.
Thus,
theψsemiprimeness
semiprimeness
implies
that
Furthermore,
we
haveThus,
ψRψthe
=
R = {0}.
{0}. Reusing
the
fact
that
is0.a0.
Furthermore,
have
have ψRψ
ψRψ =
= ψψ2222R
R=
= {0}. Reusing
Reusing the
the fact
factthat
thatR
R isaa
semiprime
ring,
it
is
concluded
that
ψ
=
0.
This
is
exactly
what
we
had
Furthermore,
we
have
ψRψ
=
ψ
Furthermore,
we
have
ψRψ
=
ψ
R
=
{0}.
Reusing
the
fact
that
R
isa a
R
=
{0}.
Reusing
the
fact
that
R
is
semiprime ring, it
it is
is concluded
concluded that
that ψψ =
= 0.
0. This
This isis exactly
exactlywhat
whatwe
wehad
had
to
prove. ring,
itit isisconcluded
semiprime
ring,
concludedthat
thatψψ==0.0.This
Thisisisexactly
exactlywhat
whatwewehad
had
semiprime
to
prove.
to prove.
prove.
to
The Previous
Previous theorem
theorem implies the
the following corollaries.
corollaries.
The
theorem implies
implies the following
following corollaries.
The Previous
Previous theorem
theorem implies
impliesthe
thefollowing
followingcorollaries.
corollaries.
The
Corollary 2.10.
2.10. Let
Let R
R be
be aa 2-torsion
2-torsion free
free semiprime
semiprime ring and
and σ, T :
Corollary
2.10. Let R be a 2-torsion 2free semiprime ring
ring and σ,σ,TT : :
R
→R
R be
be additive
additive
maps
such
that T (x22free
Corollary
2.10. Let
Let
R be
be
2-torsion
ring
and
σ,
Corollary
2.10.
R
aa 2-torsion
)free
= semiprime
Tsemiprime
(x)σ(x) for
ring
alland
x ∈σ,R.
T TIf
: :
R
→
additive maps
maps such
such that
that TT(x
(x2 2)) =
= TT(x)σ(x)
(x)σ(x) for
for all
all xx∈∈R.
R. IfIf
R→
→R
R be
be additive
additive maps
maps such
suchthat
thatTT(x
(x) )==TT(x)σ(x)
(x)σ(x)for
forallallx x∈∈R.R.IfIf
R
AAGENERAL
GENERAL
CHARACTERIZATION
CHARACTERIZATION
OF ADDITIVE
OF ADDITIVE
...
... 133
133
T (xy)σ(z)
(xy)σ(z)TT(xy)z
(xy)z
T (x)σ(yz)
T (x)σ(yz)
+ T+
(x)σ(y)z
T (x)σ(y)z
=0=
for0allforx, all
y, z x,
∈ y,
R,z ∈ R,
then
then TT(xy)
(xy)==T T(x)σ(y)
(x)σ(y)
forfor
all all
x, yx,∈yR.
∈ R.
2 ) 2=
Proof.
Proof.Note
Notethat
thatT (x
T (x
)=
T (x)σ(x)
T (x)(x
+ σ(x)
x)(x)x
= T+(x)x +
T (x)σ(x)
= T=
(x)(x
+ σ(x)
x)
=T
(x)(σ(x)x)x)forforallall
∈ R.
defining
= T (x)(σ(y)
y) and
T (x)(σ(x)
x ∈x R.
By By
defining
(x,(x,
y) =y)
T (x)(σ(y)
y) and
using the
thehypothesis,
hypothesis,
have
using
wewe
have
(xy,z)
z)(x,
(x,yz)
yz)
(x,
T (xy)(σ(z)
T (x)(σ(yz)
(xy,
++
(x,
y)zy)z
= T=(xy)(σ(z)
z)
z)
T (x)(σ(yz)
yz) yz)
T (x)(σ(y)
y)z
+ T+(x)(σ(y)
y)z
T (xy)σ(z)
T (xy)z
T (x)σ(yz)
= T=(xy)σ(z)
T (xy)z
T (x)σ(yz)
T (x)yz
T (x)σ(y)z
T (x)yz
+ T+(x)yz
+ T+
(x)σ(y)z
T (x)yz
T (xy)σ(z)
T (xy)z
T (x)σ(yz)
+ T (x)σ(y)z
= T=(xy)σ(z)
T (xy)z
T (x)σ(yz)
+ T (x)σ(y)z
= 0.= 0.
It means
meansthat
thatis isa l-semi
a l-semi
Hochschild
2-cocycle.
By using
Theorem
It
Hochschild
2-cocycle.
By using
Theorem
2.9, 2.9,
it isis obtained
obtainedthat
that
it
(xy)==T (x)y
T (x)y
+ (x,
y)T=
T (x)y
+ T (x)(σ(y)
TT(xy)
+ (x,
y) =
(x)y
+ T (x)(σ(y)
y) y)
T (x)y
T (x)σ(y)
T (x)y
==T (x)y
+ T+(x)σ(y)
T
(x)y
T (x)σ(y)
==T (x)σ(y)
for all
all x,x,yy∈∈R.R.
for
Corollary2.11.
2.11.Let
Let
a 2-torsion
semiprime
ringδ and
δ:R→
Corollary
RR
be be
a 2-torsion
free free
semiprime
ring and
:R→
2
2
R be
be aaJordan
Jordangeneralized
generalized
σ-derivation,
) = δ(x)σ(x)+σ(x)d(x)
R
σ-derivation,
i.e. i.e.
δ(x δ(x
) = δ(x)σ(x)+σ(x)d(x)
for each
eachxx∈∈RRand
andsome
some
σ-derivation
on Assume
R. Assume
for
σ-derivation
d ond R.
that Tthat
= δ T =
d δd
has the
thefollowing
followingproperty:
property:
has
(xy)σ(z)TT(xy)z
(xy)z
T (x)σ(yz)
T (x)σ(y)z
= 0 all
f orx,all
TT(xy)σ(z)
T (x)σ(yz)
+ T+
(x)σ(y)z
= 0 for
y, z x,
∈ y,
R.z ∈ R.
Then, δδisisaageneralized
generalized
σ-derivation.
Then,
σ-derivation.
Proof.We
Wehave
have
Proof.
22
2 2
2 2
δ(x
d(x
TT(x(x
) )==δ(x
) ) d(x
) )
δ(x)σ(x)
+ σ(x)d(x)
d(x)σ(x)
σ(x)d(x)
==δ(x)σ(x)
+ σ(x)d(x)
d(x)σ(x)
σ(x)d(x)
(δ(x)
d(x))σ(x)
==(δ(x)
d(x))σ(x)
T (x)σ(x)
==T (x)σ(x)
134
A. HOSEINI
for all x ∈ R. According to the previous corollary, T (xy) = T (x)σ(y)
for all x, y ∈ R. Hence,
δ(xy) = T (xy) + d(xy) = T (x)σ(y) + d(x)σ(y) + σ(x)d(y)
= δ(x)σ(y) + σ(x)d(y)
for all x, y ∈ R.
The following corollary has been proved by Vukman in [10]. Now, we
present an alternative proof for it.
Corollary 2.12. Let R be a 2-torsion free semiprime ring and
δ : R → R be a Jordan generalized derivation. In this case δ is a
generalized derivation.
Proof. We have the relation δ(x2 ) = δ(x)x + xd(x) for all x ∈ R, where
d is a Jordan derivation of R. It follows from Theorem 1 of [4] that d is
a derivation. The proof is completed by substituting σ = id in Corollary
2.11.
Acknowledgements
The author is greatly indebted to the referee(s) for his/her valuable
suggestions and careful reading of the paper.
References
[1] S. Ali and C. Haetinger, Jordan -centralizers in rings and some applications, Bol. Soc. Paran. Mat., 26 (2008), 71-80.
[2] M. Ashraf, N. Rehman, Sh. Ali, and M. R. Mozumder, On generalized
(θ, φ)-derivations in semiprime rings with involution, Math. Slovaca, 62
(3) (2012), 451-460.
[3] M. Ashraf, On left (θ, ϕ)-derivations of prime rings, Archivum Mathematicum, 41 (2) (2005), 157-166.
[4] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math.
Soc, 140 (4) (1988), 1003-1006.
A GENERAL CHARACTERIZATION OF ADDITIVE ...
135
[5] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc, 3 (1988), 321-322.
[6] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc., 53 (1975),
1104-1110.
[7] G. Dales, et al., Introduction to Banach Algebras, Operators and Harmonic Analysis, Cambridge University press, 2002.
[8] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc.,
8 (1957), 1104-1110.
[9] J. Vukman and I. Kosi-Ulbl, Centralizers on rings and algebras, Bull.
Austral. Math. Soc., 71 (2005), 225-234.
[10] J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese Journal of Mathematics, 11 (2) (2007), 367-370.
[11] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin, 32 (4) (1991), 609-614.
Amin Hosseini
Department of Mathematics
Assistant Professor of Mathematics
Kashmar Higher Education Institute
Kashmar, Iran
E-mail: A.Hosseini@mshdiau.ac.ir