Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Some results On Semiderivations Of Semiprime
Semirings
Abstract- Let S be a semiprime semiring
...
In this paper we try to generalize some properties of prime rings
with derivations to semiprime semirings with semiderivations
...

I
...
For any x, y  S ,
respectively
...
C
...
He obtained some results of derivations of prime rings
into semiderivations
...
E
...
S
...
C
...
Chuang 8 studied on the structure of semiderivations in prime rings
...
J
...
Grezesczuk 5 obtained
the commutativity properties of semiprime rings with the help of semi derivations
...
Firat 3 generalized some
results of prime rings with derivations to the prime rings with semiderivations
...

II
...
1
A semiring S is a nonempty set S equipped with two binary operations  and
1
...


S , is a monoid with identity element

 such that

0

1

3
...

Definition 2
...

Definition 2
...


Definition 2
...


Definition 2
...


Definition 2
...


Definition 2
...

If g  I , ie an identity mapping of S then all semiderivations associated with g are merely ordinary derivations
...


Example2
...
Let S  S1  S 2
...
Define f : S  S such that
f x1 , x2   0,  2 x2  and g : S  S such that g x1 , x2   1 x1 ,0 for all x1  S1 , x2  S 2
...
Then it can easily be seen that f is a semiderivation of S (with associated

addition and multiplication on S by

map g) which is not a derivation
...
RESULTS
Lemma 3
...
If S admits a semiderivation f such that
af x  0 or f xa  0 for all x  S then a  0 or f  0
...
Hence a  0 or f  0
...
2
Let S be a semiprime semiring, f a nonzero semiderivation of S associated with a function g( not necessarily
surjective)
...

Proof:

For any x, y, z  S
f  x y  z   f  x g  y  z   xf  y  z 
 f  x g  y  z   xf  y   xf  z 

(1)

Also for any x, y, z  S
f x y  z   f  xy  xz
 f  xy  f xz
 f  x g  y   xf  y   f x g z   xf z 

(2)

Comparing (1) and (2)

g  y  z   g  y   g z  for all y, z  S
Now for any x, y, z  S
f xyz   f xyg  z   xyf  z 
 f  x g  y g  z   xf  y g  z   xyf  z 

(3)

Also ,
f  xyz   f  x yz 
 f  x g  yz   xf  yz 
 f x g  yz   xf  y g  z   xyf z 

(4)

Comparing (3) and (4)

g  yz   g  y g z  for all y, z  S
Hence g is a homomorphism of S
...
3
Let S be a semiprime semiring, f a semiderivation of S such that

f S   Z then f  0 or S is commutative
...
Since S is prime

f x  0 for all x  S or y, x  0 for all x, y  S
...


Lemma 3
...

Proof:
By hypothesis

f 2 x   0 for all x  S

Replace x by xy

xy  0 , for all x, y  S
0  f  f xy, for all x, y  S
 f  f x g  y   xf  y , for all x, y  S
 f 2 x g g  y   f x  f g  y   f x g  f  y   xf 2  y , for all x, y  S
 2 f x  f g  y , for all x, y  S
f

2

Since S is 2- torsion free and g is surjective we have

f x f  y   0 for all x, y  S

Replace y by yz

4

f  x  f  yz   0 for all x, y, z  S
f  x  f  y g z   f x  yf  z   0 for all x, y, z  S
f  x  yf z   0 for all x, y, z  S
f  x sf z   0 for all x, z  S
Since S is prime

f x  0 orf z   0 all x, z  S

In both the cases f  0
...
5
Let S be a 2-torsion free semiprime semiring and a is an element in S
...

Proof:
By hypothesis f ( x), a  0 for all x  S
Replace x by xy









 f ( xy), a  0 for all x, y  S
0   f ( x) g ( y )  xf ( y ), a 
  f ( x) g ( y ), a   xf ( y ), a 
 f ( x)g ( y ), a    f ( x), a g ( y )  x f ( y ), a   x, a  f ( y )
 f ( x)g ( y ), a   x, a  f ( y ) for all x, y  S
Since g is surjective, 0  f ( x)y, a  x, a f ( y) for all x, y  S
Replace y by f  y 

0  f ( x) f ( y), a  x, a f 2 ( y) for all x, y  S

 x, a f 2 ( y) for all x, y  S
Replace x by xz
0  xz, a  f 2 ( y ) for all x, y, z  S

 xz , a  f 2 ( y )  x, a zf 2 ( y ) for all x, y, z  S
 x, a zf 2 ( y ) for all x, y, z  S
 x, a sf 2 ( y ) for all x, y  S

Since S is Prime,

x, a  0 or f 2 ( y)  0
x, a  0  a  Z (S ) and f 2 ( y)  0 

f  0 by lemma 3
...
6
Let S be a 2-torsion free semiprime semiring and f a semiderivation of S such that
or S is commutative
...
4 and 3
...


Theorem 3
...


Proof:
By hypothesis

 f ( x), x  0 for all x  S

Linearizing,

0   f ( x), y   f ( y), x
Replacing

y by yx

6

 f ( x), x  0 for all x  S

0   f ( x), yx    f ( yx), x  for all x, y  S
  f ( x), y x  y f ( x), x    f ( y ) x  g ( y ) f ( x), x 
  f ( x), y x  f ( y )x, x    f ( y ), x x  g ( y ) f ( x), x   g ( y ), x  f ( x)
 g ( y ), x  f ( x)
Since g is surjective

0   y, x  f ( x) for all x, y  S
Re place y by yz
0   yz , x  f ( x) for all x, y, z  S
 yz , x  f ( x)   y, x zf ( x) for all x, y, z  S
  y, x zf ( x) for all x, y, z  S
  y, x sf ( x) for all x, y  S
Since S is prime

y, x  0 or f ( x)  0

This means that S is commutative or

f  0
...
8
Let S be a semiprime semiring, f a nonzero semiderivation of S such that
is commutative
Proof:
By hypothesis
Replacing

f x, y  0 for all x, y  S

y by xy

0  f x, xy for all x, y  S
 f xx, y  for all x, y  S
 f ( x) g x, y   xf x, y  for all x, y  S
 f ( x) g x, y  for all x, y  S
 f ( x)g ( x), g ( y )sin ce g is a hom omorphism
 f ( x)x, y sin ce g is surjective
Re place y by yz
0  f ( x)x, yz  for all x, y, z  S

 f ( x) yx, z  for all x, y, z  S
 f ( x) sx, z  for all x, z  S

Since S is prime ,

7

f x, y  0 for all x, y  S
...

Theorem 3
...
Then S is commutative
...
8 S is commutative
...
10
Let S be a 2- torsion free semiprime semiring and f is a semiderivation of S with
endomorphism
...

Proof:
By hypothesis,

af x, x, x  0 for all x  S

Linearising

af x, x, y  af x, y, x  0 for all x, y S
Replacing

g : S  S is an onto

y by yx

af x, x, yx  af x, yx, x  0 for all x, y S

8

0  af x , x , yx   af x , yx , x  for all x, y  S
 af x , x , y x  yaf x , x , x   af x , y x, x    yaf x , x, x 
 af x , x , y x  af x , y x, x   af x , y , x x  yaf x , x, x    y, x af x , x 
  y, x af x , x  for all x, y  S
Re place y by zy
0  zy , x af x , x  for all x, y, z  S
 z y, x af x , x   z, x yaf x , x 
 z, x yaf x , x  for all x, y, z  S
In particular

0  af  x , x yaf  x , x  for all x, y  S
 af  x , x saf  x , x  for all x  S
By semiprimeness of S ,

af x, x  0 for all x S

Hence x  af x is commuting on S
...
11
Let S be a non commutative 2- torsion free semiprime semiring and f is a semiderivation of S with
onto endomorphism
...


Proof:
By hypothesis,

af x, x, x  0 for allx  S

Then by lemma 3
...


REFERENCES
1
...
Bresar, On the distance of the compositions of two derivations to the generalized derivations,Glasgow J
...
, 33(1), (1991), 89-93
...
Oznur Golbasiand Onur Agirtici, On Semiderivations of *−prime rings, Bol
...
Paran
...
,(2015),177-184
...
Alev Firat, Some Results For Semi derivations Of Prime Rings, International Journal of Pure and Applied Mathematics
...

4
...
Bangladesh Math
...
, (2011), 65-70
5
...
, and Grzesczuk,P
...
London Math Soc
...

6
...
C
...
Math
...

7
...
E and Martindale,W
...
III, Semiderivations and commutativity in prime rings,Canad
...
Bull
...

8
...
Amer
...
Soc
...


10



---