Research Journal of Applied Sciences, Engineering and Technology 6(22): 4129-4137, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: November 19, 2012
Accepted: May 07, 2013
Published: December 05, 2013
The θ-Centralizers of Semiprime Gamma Rings
1
1
M.F. Hoque, 1H.O. Roshid and 2A.C. Paul
Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh
2
Department of Mathematics, Rajshahi University, Rajshahi-6205, Bangladesh
Abstract: Let M be a 2-torsion free semiprime Γ-ring satisfying a certain assumption and θ be an endomorphism of
M. Let T : M  M be an additive mapping such that T ( xy x )   ( x )T ( y )  ( x ) holds for all x, y M and
 ,    . Then we prove that T is a θ-centralizer. We also show that T is a θ-centralizer if M contains a
multiplicative identity 1. 2010 Mathematics Subject Classification, Primary 16N60, Secondary 16W25, 16U80.
Keywords: Centralizer, θ-centralizer, Jordan centralizer, Jordan θ-centralizer, left centralizer, left θ-centralizer,
semiprime Γ-ring,
INTRODUCTION
The notion of a Γ -ring was first introduced as an
extensive generalization of the concept of a classical
ring. From its first appearance, the extensions and
generalizations of various important results in the
theory of classical rings to the theory of Γ-rings have
been attracted a wider attentions as an emerging field of
research to the modern algebraists to enrich the world
of algebra. Many prominent mathematicians have
worked out on this interesting area of research to
determine many basic properties of Γ-rings and have
extended numerous remarkable results in this context in
the last few decades. All over the world, many
researchers are recently engaged to execute more
productive and creative results of Γ-rings. We begin
with the definition.
Let M and Γ be additive abelian groups. If there
exists a mapping ( x, , y)  xy of MM M ,
which satisfies the conditions:

( xy )  M
( x  y )z  xz  yz , x(   ) z  xz  xz ,

( x α y ) β z = x α ( y β z ) for

x ( y  z )  xy  xz
all
and  ,    , then M is called a
Let M be a Γ -ring. Then an additive subgroup U of
M is called a left (right) ideal of M if
MU  U (UM  U ) . If U is a left and a right
ideal, then we say U is an ideal of M . Suppose again
that M is a Γ.-ring. Then M is said to be a 2-torsion free
if
2x  0 implies x  0 for all x  M . An ideal P1
of a Γ -ring M is said to be a prime if for any ideals A
and B of M, AB  P1 implies A  P1 or B  P1 .
An ideal P2 of a Γ-ring M is said to be semiprime if for
any ideal U of M, UU  P2 implies U  P2 . A Γ-ring
M is said to be prime if a  M  b  ( 0 ) with,
a, b  M implies a  0 or b  0 and semiprime if
a  0.
implies
aMa  (0) with a  M
Furthermore, M is said to be commutative Γ-ring if
xy  yx for all x, y  M and    . Moreover,
the set Z(M) {x M : xy  yx for all
If
M is a  -ring,
then [ x, y ]  x y  y x is
 , where x, y  M and
commutator identities:
 -ring.
Every ring M is a Γ-ring with M = Γ. However a Γ
-ring need not be a ring. Gamma rings, more general
than rings, were introduced by Nobusawa (1964).
Bernes (1966) weakened slightly the conditions in the
definition of Γ-ring in the sense of Nobusawa (1964).
and
y  M } is called the centre of the Γ-ring M.
known as the commutator of x and
x, y , z  M
 
y
with respect to
   . We make the basic
[ xy, z]  [ x, z] y  x[ ,  ] z y  x[ y, z] and
[ x , y  z ]   [ x , y ]   z  y[ ,  ] x z  y [ x , z ]  for
all x , y , z  M and  ,    . We consider the
following assumption:
Corresponding Author: M.F. Hoque, Department of Mathematics, Pabna University of Science and Technology, Pabna-6600,
Bangladesh
4129
Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
 x y z  x y z …………..
x, y , z  M and  ,    .
(A)
for
all
According to the assumption (A), the above two
identities reduce to [ xy, z ]   [ x, z ]  y  x [ y, z ] 
and
[ x, yz ]   [ x, y ]   z  y [ x, z ]  ,which we
extensively used.
An additive mapping T : M  M is a left (right)
centralizer if T ( xy )  T ( x)y[T ( xy )  xT ( y )]
holds for all x, y, z  M and  ,    . A centralizer
is an additive mapping which is both a left and a right
centralizer. For any fixed and Г,, the
mapping
is a left centralizer and
is a right centralizer. We shall restrict our
attention on left centralizer, since all results of right
centralizers are the same as left centralizers. An
additive mapping ∶ ⟶ is called a derivation if
holds
for
all
, Гand is called a Jordan derivation if
for all and Г.
An additive mapping ∶ ⟶ is Jordan left (right)
centralizer if
for all and Г.
Every left centralizer is a Jordan left centralizer but
the converse is not in general true.
An additive mappings
∶ ⟶
is called a
Jordan centralizer if
, for all , and Г. Every centralizer is
a Jordan centralizer but Jordan centralizer is not in
general a centralizer.
Bernes (1966), Luh (1969) and Kyuno (1978)
studied the structure ofГ -rings and obtained various
generalizations of corresponding parts in ring theory.
Zalar (1991) worked on centralizers of semiprime
rings and proved that Jordan centralizers and
centralizers of this rings coincide. Vukman (1997,
1999, 2001) developed some remarkable results using
centralizers on prime and semiprime rings.
Ceven (2002) worked on Jordan left derivations on
completely prime Г -rings. He investigated the
existence of a nonzero Jordan left derivation on a
completely prime Г-ring that makes the Г-ring
commutative with an assumption. With the same
assumption, he showed that every Jordan left derivation
on a completely prime Г-ring is a left derivation on it.
The commutativity condition of a prime ring has been
studied by Mayne (1984) by means of a Lie ideal of R
and with a nontrivial automorphism or derivation. Ullah
and Chaudhary (2010) proved that if T is an additive
mapping of a 2-torsion free semiprime ring with
involution * satisfying T (xx* )  T (x) (x* )   (x* )T (x) , then
t is a -centralizer.
Hoque and Paul (2011) have proved that every
Jordan centralizer of a 2-torsion free semiprime Г -ring
satisfying a centain assumption is a centralizer.
Hoque and Paul (2012) have proved that if T is an
additive mapping on a 2-torsion free semiprimeГ -ring
M with a certain assumption such that
, for all , and , Г, then is a
centralizer.
Ullah and Chaudhary (2012), have proved that
every Jordan -centralizer of a 2-torsion free semiprime
Г -ring is a -centralizer. Hoque et al. (2012) proved
that T is a
-centralizer by using a relation
2T (aba)  T (a)(b)(a)  (a)(b)T (a) , where
T is an additive mapping on a 2-torsion free
semiprimeГ -ringM.
Our research works inspired by the works of
Hoque et al. (2012) in Г -rings with -centralizers. Here
we prove that if is a 2-torsion free semiprime Г-ring
satisfying the assumption (A) and if ∶ ⟶ is an
additive mapping such that:
(1)
For all , , , Гand an endomorphism
on , then is -centralizer. Also we show that is a
centralizer if contains a multiplicative identity 1.
THE -CENTRALIZER OF SEMIPRIME
GAMMA RINGS
In this section, we have given the following
definitions:
Let be a 2-torsion free semiprime Г -ring and let
be an endomorphism of . An additive mapping
∶ ⟶ is a left (right) -centralizer if
holds for all
, and Г. If t is a left and a right -centralizer,
then it is natural to call a -centralizer.
Let be a Г-ring and let and Гbe fixed
element. Let ∶ ⟶ be an endomorphism. Define
a mapping
∶ ⟶ by
. Then it is
clear that is a left -centralizer. If
is
defined, then is a right -centralizer.
An additive mapping ∶ ⟶
is a Jordan left
(right) -centralizer if
=
holds for all and Г.
It is obvious that every left -centralizer is a Jordan
left -centralizer but in general Jordan left -centralizer
is not a left -centralizer.
Example 2.1 Let
be a Г-ring and let ∶ ⟶ be
a left
-centralizer, where
∶ ⟶ is an
endomorphism.
Define
, ∶ and
Γ
, ∶ Γ . The addition and multiplication
on
are defined as follows:
,
,
,
and
,
,
,
,
.
4130 Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
Under these addition and multiplication, it is clear
that is a Γ -ring.
,
Define
,
,
and
,
. Then T is an additive mapping on and
is an endomorphism on
. Now, let
, ,
, ,then we have:
,
,
,
,
,
,
,
,
,
,
,
.
,
,
,
,
,
,
Example 2: Let
be a Γ-ring satisfying the
assumption (A) and let be a fixed element of such
that , the centre of . Define a mapping
∶ ⟶ by
,
where
∶ ⟶
is an endomorphism and Гis a fixed
element.Then for all , and Г , we have:
,
,
Therefore, T is a Jordan left -centralizer which is
not a left -centralizer.
Let
be a Γ-ring and let be an endomorphism
on . An additive mapping ∶ ⟶
is called a
Jordan -centralizer if
, for all , and Г. It is clear that
every -centralizer is a Jordan -centralizer but the
converse is not in general a -centralizer.
An additive mapping
∶ ⟶
is a
, derivation if
holds for all , and Гand is called a Jordan
, -derivation
if
,
holds for all and Г.Now we
begin with two examples which are ensure that a centralizer and a Jordan -centralizer exist in Γ-ring.
Also,
,
,
. Under these addition and
multiplication
is a Γ -ring.Define
,
,
and
,
,
.Then is
is an endomorphism on
.
additive mapping and
Let
, ,
, ,
, . Then we have:
, ,
,
,
,
,
,
,
,
,
,
,
,
,
,
Therefore, is a Jordan -centralizer on
which is not a -centralizer.
For proving our main results, we need the
following Lemmas:
Lemma 1: (7, Lemma 1) Suppose is a semiprime Γring satisfying the assumption
. Suppose that the
relation
0 holds for all ∈ , some
, , ∈ and , ∈ Γ. Then
0 is
satisfied for all ∈ and , ∈ Γ.
Lemma 2: Let be a 2-torsion free semiprimeΓ –ring
satisfying the assumption
and let ∶ →
be
an additive mapping. Suppose that
holds for all , ∈
and , ∈ Γ.
Then:
,
,
,
,
,
,
0
0.
0
Proof: We prove (i)
T x , x  , x 
0
.
Therefore, is a left and right
is a
.
0
0
For linearization, we put
relation(1),we obtain:
centralizer, so
Example 3: Let be a Γ-ring and let ∶ → be a
-centralizer, where ∶ → is an endomorphism.
, : ∈
and Γ , : ∈Γ .
Define
as follows:
Define addition and multiplication on
,
,
,
and
,
,
(2)
xz
for x in
T  x  y  z  z  y  x     x  T  y   z     z  T  y   x 
Replacing y for x and z for y in (3), we have:
(3)
T  x  x  y  y  x  x     x  T  x   x     y  T  x   x 
4131 (4)
Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
For z   x  x in relation (3) reduces to:
,
2


2





2



T x  xyx   x  xyx    x T  xxy  yxx   x 
,
,
,
,
0


,
follows, i.e.,
0
(12)
,
,
,
,
,
,
,
(8)
Putting
,
,
,
,
0
in the above relation, we have:
,
Using Lemma 2 in the above relation , we have:
,
,
,
,
 x   y  T x   x  x     x   y  x 
T  x  x    x  T x ,  x    y  x   0
,
,
,
,
,
,
,
0
Adding the above two relations, we have:
,
,
2
2
0
,
,
0
,
0
,
,
2
0
,
,
,
,
,
0
0
9 ,
,
,
(13)
0
,
,
(9)
,
,
,
,
,
,
Putting
for y in (13) and using (1), (2), (13)
and assumption (A), we have:
0
,
,
,
Since M is 2-torsion free semiprime Г-ring, so, we
have:
0
,
,
Let
The linearization of (2) gives:
 y  x    x 
  y  T  x ,  x   x   0
2
Since
is semiprime, so
,
0
,
Now, we prove the relation
∶
Combining (6) and (7), we arrive at:
T  x  T  x ,   x  
,
,
T  x  x y x  x y  x  x
2
   x  T  x   y   x     x   y  T  x   x   x 
(7)
2
,
(11)
which gives because of (4):
2
gives:
Subtracting 11 from 10 one obtains:
The substitution xx y  yxx for y in the
relation (1) gives:
2
,
0
(6)
2
(10)
Right multiplication of 9 by
T  x  xyx  x y  x 
  xT xx yx   x yxT x x
2
,
0
T x y  x   x   x  x y x
2
2
   x  T  y   x     x     x     x  T  y   x 
(5)
Putting y  xyx in (4), we obtain:
2
,
,
4132 ,
Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
,
[T x , x  x  x ]  T x , x   x 
 x   y  T x , x   0
,
,
,
,
,
,
,
,
,
which reduces to:
 x  T x , x   x    x  x 
,
 T x ,  x    y  T x ,  x 
,
,
,
,
,
Relation
\
,
,
makes it possible to write
instead of   x  T  x ,   x  ,
which means that   x   x  T  x ,   x  can be
,
replaced by   x  T  x ,   x    x  in the above
relation. Thus we have:
  x  T  x ,   x   x   y  T  x ,   y   0
Right multiplication of the above relation by
and substitution
for
gives
finally,
,
0
,
(2)
T x , x   x 
,
,
T x , θ x βθ y β\θ x γθ x θ x γθ x βθ y β T x , θ x
θ x γθ y βθ x β T x , θ x _α
T x , θ x _αβθ x βθ y γθ x . Therefore, we have:
,
 0 0 Hence
For all x,y ∈ , ∈ , which reduces because of (3)
and (8) to:
Left multiplication of the above relation by xβ gives:  x  T x ,  x    y  x  x   y  T x , x   0 One can replace in the above relation
according to (15), θ(x)β[T(x), θ(x)]α βθ(y)βθ(x) by
  x   y  T  x ,   x    x  which gives:
  x   y  T x , x   x  x 
   x   x   x   y  T  x ,   x   0
by
of
M,
we
,
0,
∈ ,
First, we putting
because of (12):

∈
for
(17)
in (8), gives
,
(14)
0 (18)
for
in (25), we have:
,
αθ
0 (19)
,
Right multiplication of (18) by
,
(15)
have
Next we prove the relation (iii):
T  x   x   y  T  x ,   x    x   x 
0 (20)
Subtracting (20) from (19), we have:
The substitution T(x)αθ(y) for y in (14), we have:
,
0
 x  T x   y  T x ,   x   x  x 
⇒
   x   x   x T  x   y  T  x ,   x   0 (16)
,
,
0
⇒
Subtracting (16) from (15), we obtain:
T x , x    y  T x , x   x  x 
 T  x ,   x   x   x    y  T  x ,   x 
semiprimeness
0.
The substitution
Left multiplication of the above relation by T(x)α gives:
 T  x   x   x   x   y  T  x , x   0
,
⇒
0 From the above relation and Lemma-2.1, it follows that:
According
,
gives:
4133 ,
,
,
,
to
,
0
,
0
(2),
by
one
can
,
replaced
which
Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
,
0, ,
∈
, ,
∈

 ( x ) (T ( xy  yx )  T ( y ) ( x )
,
  ( x )T ( y ))  ( x)  0.
Hence by semiprimenessof M,  ( x) [T ( x), ( x)]  0
,
We define:
Finally, we prove the equation (u) :
G ( ( x ),  ( y ))  T ( xy  yx )  T ( y ) ( x )   ( x )T ( y ).
x, y  M ,  ,   .
T ( x ),  ( x) = 0
Then it is clear that:
(21)
 ( x )G ( ( x ),  ( y ))  ( x)  0 and
G ( ( x ),  ( y ))  G ( ( y ),  ( x )).
From (2) and (17), it follows that:
[T ( x ),  ( x)]  ( x) = 0, x  M ,  ,   .
Replacing x for y and using (22), we have:
The linearization of the above relation gives (see
how relation (13) was obtained from (2)):
 ( y )G ( ( x), ( y )) ( y )  0. We can also easily
prove the following results : 
[T ( x ),  ( x )]  ( y )  [T ( x ),  ( y )]
(1) G ( ( x)   ( z ), ( y ))  G ( ( x), ( y ))
 G ( ( z), ( y))
(2) G ( (x), ( y)   (z))  G ( (x), ( y))  G ( ( x), ( z ))
 ( x )  [T ( y ),  ( x )]  ( x )  0
(3) G   ( ( x), ( y ))  G ( ( x), ( y ))
 G  ( ( x), ( y ))
(4) G ( ( x), ( y))  G ( ( x), ( y))
(5) G ( ( x), ( y ))  G ( ( x), ( y )) .
Right multiplication of the above relation by
 [T ( x), x] gives because of (17):
[T ( x), ( x)]  ( y )  [T ( x), ( x)]  0
which implies [T ( x ),  ( x )]  0.
Lemma 3: Let M be  - ring satisfying the
assumption (A) and let T : M  M be an additive
mapping such that T ( xyx)   ( x)T ( y )  ( x)
holds for all x, y, M and  ,   . Then:
(a) [G ( ( x),  ( y )),  ( x)]  0
 ( x) (T ( xy  yx)  T ( y) ( x)   ( x)T ( y)) ( x)  0
(22)
Proof: The substitution
gives:
xy  yx
for
Lemma 4: Let M be a 2-torsion free semiprime Γring satisfying the assumption (A) and let T : M  M
be
an
additive
mapping.
Suppose
that
T ( xyx)   ( x)T ( y)  ( x) holds for all x, y  M
and  ,    . Then:
(b) Gα(θ(x),θ)) = 0
Proof: First we prove the relation (a):
y in (1)
T ( xxyx  xyxx)   ( x)T ( xy  yx)  ( x)
[G ( ( x), ( y)), ( x)]  0
The linearization of (21) gives:
(23)
On the other hand, we obtain by putting z  xx
in (3), we have:
T ( xxy x  xy xx )   ( x)T ( y ) ( x ) 
 ( x)   ( x) ( x)T ( y )  ( x)
T ( xxyx  xyxx)   ( x)T ( y )
 ( x)  ( x)   ( x) ( x)T ( y )  ( x) (24)
By comparing (23) and ( 24), we have:
(25)
[T ( x),  ( y )]  [T ( y ),  ( x)]  0
(26)
Putting xy  yx for y in the above relation
and using (21), we obtain:
[T ( x), ( x) ( y)   ( y) ( x)]  [T ( xy  yx), ( x)]  0
  xT  x,  y  T x,  y  x  T xy  yx,  x  0
 T xy  yx, x   x T x,  y  T x,  y x  0
4134 Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
According to (26) one can replace T  x ,   y  by
 T  y , x 
which gives G  ( ( x ),  ( y ))  0 i.e.,
in the above relation. We have
T ( xy  yx)  T ( y) ( x)   ( x)T ( y)
therefore:
T xy  yx , x    x  T  y , x 
 T  y ,  x   x   0
i.e., G   x ,   y ,   x   0
Hence the proof is complete.
Theorem 1: Let M be a 2-torsion free semiprime Γ-ring
satisfying the assumption (A) and let T : M  M be an
additive mapping. Suppose that T(xαxβx) =
θ(x)αT(x)βθ(x) holds for all x  M and  ,    .
The proof is therefore complete.
Finally, we prove the relation (b):
Then T is a  -centralizer.
G   x ,   y   0
Proof: In particular for
reduces to:
(27)
From (22) one obtains (see how (13) was obtained
from (2)), we have:
 x G  x ,   y  z    x G
 z ,   y  x    z G  x ,   y   0
y  x,
the relation (32)
2T ( x x )  T ( x ) ( x )   ( x ) T ( x )
Combining the above relation with (21), we arrive at:
Right multiplication of the above relation by
G   x ,  y   x  gives because of (22):
  x  G   x ,   y   z  G    x ,   y   x   0
2T ( xx)  2T ( x) ( x),
xM ,  
2T ( xx)  2 ( x)T ( x),
x  M ,   .
And
Since
(28)
Relation (25) makes it possible to replace in (28),
 x G  x ,  y  by G   x ,  y   x  . Thus
we have:
M
is 2-torsion free, so we have:
T ( xx)  T ( x) ( x),
T  xx    xT x,
x  M,  
x  M ,  
G  x ,   y  x  z  G   x ,   y  x   0
(29)
By theorem 3.5 in Ullah and Chaudhary (2012), it
follows that T is a left and also right θ–centralizer
which completes the proof of the theorem.
Putting y = x in relation (1), we obtain:
Therefore by semiprimeness of M:
T  x  x  x     x  T  x   x , x  M ,  ,   T (33)
G   x ,   y   x   0
The question arises whether in a2-torsion free
semi-prime Γ–ring the above relation implies that T is a
θ-centralizer. Unfortunately we were unable to answer
affirmative if M has an identity element.
(30)
Of course, we have also:
 ( x )G ( ( x),  ( y ))  0
Theorem 2: Let M be a 2-torsion free semi-prime Γ–
ring with identity element 1 satisfying the assumption
(A) and let T : M  M be an additive mapping.
Suppose that T  xxx    xT  x x holds for
The linearization of (30) with respect to x gives:
G ( ( x),  ( y)) ( z)  G ( ( z),  ( y)) ( x)  0
all x  M and  ,    . Then T is a
Right multiplication of the above relation by
G ( ( x),  ( y)) gives because of (31):
 -centralizer.
Proof: Putting x+1 for x in relation (33), one obtains
after some calculations:
3T  xx   2T  x   T  x   x   x T  x 
   x a  x   a  x     x a
G ( ( x),  ( y )) ( z )G ( ( x),  ( y ))  0
4135 Res. J. Appl. Sci. Eng. Technol., 6(22): 4129-4137, 2013
where a stands for T(1). Putting –x for x in the relation
above and comparing the relation so obtain with the
above relation we have:
6T xx   2T  x   x   2  x T x   2 x a x 
(34)
and θ be an endomorphism of M. Suppose there exists
an additive mapping T : M  M
such that
m
n
m
n
holds
for all
T (( x ) ( x ) x)   ( x ) T ( x) (  x )
x  M ,  ,    , where m  1, n  1 are some integers.
Thus T is a θ-centralizer.
CONCLUSION
and
2T  x   a x    x 
(35)
We shall prove that a  Z M  .According to (35)
one can replace 2T  x  on the right side of (41) by
a x     x  and 6T xx  on the left side by
3a x  x   3 x  x a , which gives after some
calculation:
In this study, we have given some examples which
have shown that θ-centralizer exists in Γ-rings. We
proved that if T is an additive mapping on a 2-torsion
free semiprime Γ-ring M satisfying the assumption (A)
such that T ( x  y  x )   ( x ) T ( y )  ( x ) for all
x, y  M ,  ,    and θ an endomorphism on M,then
T is a θ-centralizer. We have also showed that T is a θcentralizer if M contains a multiplicative identity 1.
a x  x     x   x a  2  x a  x   0
REFERENCES
The above relation can be written in the form:
[[ a , ( x )] ,  ( x )]   0; x  M ,  ,   
(36)
The linearization of the above relation gives:
[[ a,  ( x )] , ( y )]   [[ a, ( y )] , ( x )]   0
(37)
Putting y  x y in (37), we obtain because of
(36) and (37):
0  [[a,  ( x)] ,  ( x) ( y)]  [[ ,  ( x) ( y)] ,  ( x)]
 [[a, ( x)] , ( x)]  ( y)]   ( x)
)[[a, ( x)], ( y)]  [[a, ( x)]  ( y)
  ( x)[a, ( y)] , ( x)]
  ( x)[[a, ( x)] , ( y)]  [[a, ( x)] 
 ( y), ( x)]  [ ( x)[a, ( y)] , ( x)]
  ( x) [[a, ( x)] , ( y)]  [[a, ( x)
 ( x)]  ( y)  [a, ( x)]  [ ( y), ( x)]
 [ ( x) [a, ( y)] , ( x)]
 [a, ( x)]  [ ( y), ( x)]
The substitution  ( y ) a for  ( y) in the above
relation gives:
[ a, ( x )]  ( y )  [a, ( x)]  0
Hence it follows a  Z (M ) , which reduces (35) to
the form T ( x)  a ( x), x  M ,  . The proof of
the theorem is complete.
We conclude with the following conjecture: let M
be a semiprime Γ-ring with suitable torsion restrictions
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