APM  Vol.8 No.2 , February 2018
The Commutativity of a *-Ring with Generalized Left *-α-Derivation
ABSTRACT
In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

1. Introduction

Let R be an associative ring with center Z ( R ) . x y + y x where x , y R is denoted by ( x , y ) and x y y x where x , y R is denoted by [ x , y ] which holds some properties: [ x y , z ] = x [ y , z ] + [ x , z ] y and [ x , y z ] = [ x , y ] z + y [ x , z ] . An additive mapping α which holds α ( x y ) = α ( x ) α ( y ) for all x , y R is called a homomorphism of R. An additive mapping β which holds β ( x y ) = β ( y ) β ( x ) for all x , y R is called an anti-homomorphism of R. A homomorphism of R is called an epimorphism if it is surjective. A ring R is called a prime if a R b = ( 0 ) implies that either a = 0 or b = 0 for fixed a , b R . In private, if b = a , it implies that R is a semiprime ring. An additive mapping : R R which holds ( x y ) = y x and ( x ) = x for all x , y R is called an involution of R. A ring R which is equipped with an involution * is called a *-ring. A *-ring R is called a prime *-ring (resp. semiprime *-ring) if R is prime (resp. semiprime). A ring R is called a *-prime ring if a R b = a R b = ( 0 ) implies that either a = 0 or b = 0 for fixed a , b R .

Notations of left *-derivation and generalized left *-derivation were given in a b u : Let R be a *-ring. An additive mapping d : R R is called a left *-derivation if d ( x y ) = x d ( y ) + y d ( x ) holds for all x , y R . An additive mapping F : R R is called a generalized left *-derivation if there exists a left *-derivation d such that F ( x y ) = x F ( y ) + y d ( x ) holds for all x , y R . An additive mapping T : R R is called a right *-centralizer if T ( x y ) = x T ( y ) for all x , y R . It is clear that a generalized left *-derivation associated with zero mapping is a right *-centralizer on a *-ring.

A *-derivation on a *-ring was defined by Bresar and Vukman in [2] as follows: An additive mapping d : R R is said to be a *-derivation if d ( x y ) = d ( x ) y + x d ( y ) for all x , y R .

A generalized *-derivation on a *-ring was defined by Shakir Ali in Shakir: An additive mapping F : R R is said to be a generalized *-derivation if there exists a *-derivation d : R R such that F ( x y ) = F ( x ) y + x d ( y ) for all x , y R .

In this paper, motivated by definition of a left *-derivation and a generalized left *-derivation in [1] , it is defined that a left *-α-derivation and a generalized left *-α-derivation are as follows respectively: Let R be a *-ring and α be a homomorphism of R. An additive mapping d : R R such that d ( x y ) = x d ( y ) + α ( y ) d ( x ) for all x , y R is called a left *-α-derivation of R. An additive mapping f is called a generalized left *-α-derivation if there exists a left *-α-derivation d such that f ( x y ) = x f ( y ) + α ( y ) d ( x ) for all x , y R . Similarly, motivated by definition of a *-derivation in [2] and a generalized *-derivation in [3] , it is defined that a *-α-derivation and a generalized *-α-derivation are as follows respectively: Let R be a *-ring and α be a homomorphism of R. An additive mapping t which holds t ( x y ) = t ( x ) y + α ( x ) t ( y ) for all x , y R is called a *-α-derivation of R. An additive mapping g is called a generalized *-α-derivation if there exists a *-α-derivation t such that g ( x y ) = g ( x ) y + α ( x ) t ( y ) holds for all x , y R .

In [4] , Bell and Kappe proved that if d : R R is a derivation holds as a homomorphism or an anti-homomorphism on a nonzero right ideal of R which is a prime ring, then d = 0 . In [5] , Rehman proved that if F : R R is a nonzero generalized derivation with a nonzero derivation d : R R where R is a 2-torsion free prime ring holds as a homomorphism or an anti homomorphism on a nonzero ideal of R, then R is commutative. In [6] , Dhara proved some results when a generalized derivation acting as a homomorphism or an anti-homomorphism of a semiprime ring. In [7] , Shakir Ali showed that if G : R R is a generalized left derivation associated with a Jordan left derivation δ : R R where R is 2-torsion free prime ring and G holds as a homomorphism or an anti-homomorphism on a nonzero ideal of R, then either R is commutative or G ( x ) = x q for all x R and q Q l ( R C ) . In [1] , it is proved that if F : R R is a generalized left *-derivation associated with a left *-derivation on R where R is a prime *-ring holds as a homomorphism or an anti-homomorphism on R, then R is commutative or F is a right *-centralizer on R.

The aim of this paper is to extend the results which proved for generalized left *-derivation of R in [1] and prove the commutativity of a *-ring with generalized left *-α-derivation. Some results are given for generalized *-α-derivation.

The material in this work is a part of first author’s Master’s Thesis which is supervised by Prof. Dr. Neşet Aydin.

2. Main Results

From now on, R is a prime *-ring where : R R is an involution, α is an epimorphism on R and f : R R is a generalized left *-α-derivation associated with a left *-α-derivation d on R.

Theorem 1

1) If f is a homomorphism on R, then either R is commutative or f is a right *-centralizer on R.

2) If f is an anti-homomorphism on R, then either R is commutative or f is a right *-centralizer on R.

Proof. 1) Since f is both a homomorphism and a generalized left *-α-derivation associated with a left *-α-derivation d on R, it holds that for all x , y , z R

f ( x y z ) = f ( x ( y z ) ) = x f ( y z ) + α ( y z ) d ( x ) = x f ( y ) f ( z ) + α ( y ) α ( z ) d ( x ) .

That is, it holds for all x , y , z R

f ( x y z ) = x f ( y ) f ( z ) + α ( y ) α ( z ) d ( x ) . (1)

On the other hand, it holds that for all x , y , z R

f ( x y z ) = f ( ( x y ) z ) = f ( x y ) f ( z ) = x f ( y ) f ( z ) + α ( y ) d ( x ) f ( z ) .

So, it means that for all x , y , z R

f ( x y z ) = x f ( y ) f ( z ) + α ( y ) d ( x ) f ( z ) . (2)

Combining Equation (1) and (2), it is obtained that for all x , y , z R

x f ( y ) f ( z ) + α ( y ) α ( z ) d ( x ) = x f ( y ) f ( z ) + α ( y ) d ( x ) f ( z ) .

This yields that for all x , y , z R

α ( y ) ( α ( z ) d ( x ) d ( x ) f ( z ) ) = 0.

Replacing y by yr where r R in the last equation, it implies that

α ( y ) α ( R ) ( α ( z ) d ( x ) d ( x ) f ( z ) ) = ( 0 )

for all x , y , z R . Since α is surjective and R is prime, it follows that for all x , z R

α ( z ) d ( x ) = d ( x ) f ( z ) . (3)

Replacing x by xy where y R in the last equation, it holds that for all x , y , z R

α ( z ) x d ( y ) + α ( z ) α ( y ) d ( x ) = x d ( y ) f ( z ) + α ( y ) d ( x ) f ( z ) .

Using Equation (3) in the last equation, it implies that for all x , y , z R

[ α ( z ) , x ] d ( y ) + [ α ( z ) , α ( y ) ] d ( x ) = 0.

Since α is surjective, it holds that for all x , y , z R

[ z , x ] d ( y ) + [ z , α ( y ) ] d ( x ) = 0.

Replacing z by x in the last equation, it follows that for all x , y R

[ x , α ( y ) ] d ( x ) = 0.

Since α is a surjective, it holds that [ x , y ] d ( x ) = 0 for all x , y R . Replacing y by yz where z R in the last equation, it gets [ x , y ] z d ( x ) = 0 for all x , y , z R . So, it implies that for all x , y R

[ x , y ] R d ( x ) = ( 0 ) .

Since R is prime, it follows that [ x , y ] = 0 or d ( x ) = 0 for all x , y R . Let A = { x R | [ x , y ] = 0 , y R } and B = { x R | d ( x ) = 0 } . Both A and B are

additive subgroups of R and R is the union of A and B. But a group can not be set union of its two proper subgroups. Hence, R equals either A or B.

Assume that A = R . This means that [ x , y ] = 0 for all x , y R . Replacing x by x in the last equation, it gets that [ x , y ] = 0 for all x , y R . Therefore, R is commutative.

Assume that B = R . This means that d ( x ) = 0 for all x R . Since f is a generalized left *-α-derivation associated with d, it follows that f is a right *-centralizer on R.

2) Since f is both an anti-homomorphism and a generalized left *-α-derivation associated with a left *-α-derivation d on R, it holds that

f ( x y ) = f ( y ) f ( x ) = x f ( y ) + α ( y ) d ( x )

for all x , y R . It means that for all x , y R

f ( y ) f ( x ) = x f ( y ) + α ( y ) d ( x ) .

Replacing y by xy in the last equation and using that f is an anti-homomorphism, it follows that for all x , y R

x f ( y ) f ( x ) + α ( y ) d ( x ) f ( x ) = x f ( y ) f ( x ) + α ( x ) α ( y ) d ( x )

which implies that for all x , y R

α ( y ) d ( x ) f ( x ) = α ( x ) α ( y ) d ( x ) . (4)

Replacing y by zy where z R in the last equation, it holds that for all x , y , z R

α ( z ) α ( y ) d ( x ) f ( x ) = α ( x ) α ( z ) α ( y ) d ( x ) .

Using Equation (4) in the above equation, it gets [ α ( z ) , α ( x ) ] α ( y ) d ( x ) = 0 for all x , y , z R . Since α is surjective, it holds that [ z , α ( x ) ] y d ( x ) = 0 for all x , y , z R . That is, for all x , z R

[ z , α ( x ) ] R d ( x ) = ( 0 ) .

Since R is prime, it implies that [ z , α ( x ) ] = 0 or d ( x ) = 0 for all x , z R . Let K = { x R | [ z , α ( x ) ] = 0 , z R } and L = { x R | d ( x ) = 0 } . Both K and L are additive subgroups of R and R is the union of K and L. But a group cannot be set union of its two proper subgroups. Hence, R equals either K or L.

Assume that K = R . This means that [ z , α ( x ) ] = 0 for all x , z R . Since α is surjective, it holds that [ z , x ] = 0 for all x , z R . It follows that R is commutative.

Assume that L = R . Now, required result is obtained by applying similar techniques as used in the last paragraph of the proof of 1).

Lemma 2 If f is a nonzero homomorphism (or an anti-homomorphism) and f ( R ) Z ( R ) then R is commutative.

Proof. Let f be either a nonzero homomorphism or an anti-homomorphism of R. From Theorem 1, it implies that either R is commutative or f is a right *-centralizer on R. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. Since f ( R ) is in the center of R, it holds that [ f ( x y ) , r ] = 0 for all x , y , r R . Using that f is a right *-centralizer and f ( R ) Z ( R ) , it yields that for all x , y , r R

0 = [ f ( x y ) , r ] = [ x f ( y ) , r ] = [ x , r ] f ( y )

which follows that for all x , y , r R

[ x , r ] f ( y ) = 0.

Since f ( R ) is in the center of R, it is obtained that for all x , y , r R

[ x , r ] R f ( y ) = ( 0 ) .

Using primeness of R, it is implied that either [ x , r ] = 0 or f ( y ) = 0 for all x , y , r R . Since f is nonzero, it means that R is commutative. This is a contradiction which completes the proof.

Theorem 3 If f is a nonzero homomorphism (or an anti-homomorphism) and f ( [ x , y ] ) = 0 for all x , y R then R is commutative.

Proof. Let f be a homomorphism of R. It holds that R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. From the hypothesis, it gets that f ( [ x , y ] ) = 0 for all x , y R . Since f is a homomorphism, it holds that for all x , y R

0 = f ( [ x , y ] ) = f ( x y y x ) = f ( x ) f ( y ) f ( y ) f ( x ) = [ f ( x ) , f ( y ) ]

i.e., for all x , y R

[ f ( x ) , f ( y ) ] = 0.

Replacing x by x z in the last equation, using that f is a right *-centralizer on R and using the last equation, it holds that 0 = [ f ( x z ) , f ( y ) ] = [ x f ( z ) , f ( y ) ] = [ x , f ( y ) ] f ( z ) for x , y , z R . So, it follows that for all x , y , z R

[ x , f ( y ) ] f ( z ) = 0.

Replacing x by xr where r R and using the last equation, it holds that [ x , f ( y ) ] r f ( z ) = 0 for all x , y , z , r R . This implies that for all x , y , z R

[ x , f ( y ) ] R f ( z ) = ( 0 ) .

Using the primeness of R, it is obtained that either [ x , f ( y ) ] = 0 or f ( z ) = 0 for all x , y , z R . Since f is nonzero, it follows that f ( R ) Z ( R ) . Using Lemma 2, it is obtained that R is commutative. This is a contradiction which completes the proof.

Let f be an anti-homomorphism of R. This holds that R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. From the hypothesis, it gets that f ( [ x , y ] ) = 0 for all x , y R . Since f is an anti-homomorphism, it holds that for all x , y R

0 = f ( [ x , y ] ) = f ( x y y x ) = f ( y ) f ( x ) f ( x ) f ( y ) = [ f ( x ) , f ( y ) ]

i.e., for all x , y R

[ f ( x ) , f ( y ) ] = 0.

After here, the proof is done by the similarly way in the first case and same result is obtained.

Theorem 4 If f is a nonzero homomorphism (or an anti-homomorphism), a R and [ f ( x ) , a ] = 0 for all x R then a Z ( R ) or R is commutative.

Proof. Let f be either a homomorphism or an anti-homomorphism of R. It holds that R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. From the hypothesis, it yields that for all x , y R

0 = [ f ( x y ) , a ] = [ x f ( y ) , a ] = x [ f ( y ) , a ] + [ x , a ] f ( y ) = [ x , a ] f ( y )

i.e., for all x , y R

[ x , a ] f ( y ) = 0.

Replacing x by xr where r R , it holds that [ x , a ] r f ( y ) = 0 for all x , y , r R . This implies that [ x , a ] R f ( y ) = ( 0 ) for all x , y R . Using the primeness of R, it implies that [ x , a ] = 0 or f ( y ) = 0 for all x , y R . Since f is nonzero, it follows that a Z ( R ) . That is, it is obtained that either a Z ( R ) or R is commutative.

Theorem 5 If f is a nonzero homomorphism (or an anti-homomorphism) and f ( [ x , y ] ) Z ( R ) for all x , y R then R is commutative.

Proof. Let f be a nonzero homomorphism of R. It implies that either R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. Since f is a homomorphism and f ( [ x , y ] ) Z ( R ) for all x , y R , it holds that for all x , y R

f ( [ x , y ] ) = f ( x y y x ) = f ( x y ) f ( y x ) = f ( x ) f ( y ) f ( y ) f ( x ) = [ f ( x ) , f ( y ) ]

i.e., for all x , y R

[ f ( x ) , f ( y ) ] Z ( R ) .

It means that [ [ f ( x ) , f ( y ) ] , r ] = 0 for all x , y , r R . Replacing x by x z where z R in the last equation, it holds that for all x , y , z , r R

0 = [ f ( x z ) , f ( y ) ] , r ] = [ [ x f ( z ) , f ( y ) ] , r ] = [ x , r ] [ f ( z ) , f ( y ) ] + [ [ x , f ( y ) ] , r ] f ( z ) + [ x , f ( y ) ] [ f ( z ) , r ]

which implies that for all x , y , z , r R

[ x , r ] [ f ( z ) , f ( y ) ] + [ [ x , f ( y ) ] , r ] f ( z ) + [ x , f ( y ) ] [ f ( z ) , r ] = 0.

Replacing x by f ( y ) and r by f ( z ) , it is obtained that for all x , y , z R

[ f ( y ) , f ( z ) ] [ f ( z ) , f ( y ) ] = 0.

The last equation multiplies by r from right and using that [ f ( x ) , f ( y ) ] Z ( R ) for all x , y R , it follows that for all x , y , z , r R

[ f ( y ) , f ( z ) ] r [ f ( z ) , f ( y ) ] = 0

i.e., for all x , y , z , r R .

[ f ( z ) , f ( y ) ] R [ f ( z ) , f ( y ) ] = ( 0 ) .

Using primeness of R, it is implied that for all y , z R

[ f ( z ) , f ( y ) ] = 0.

From Theorem 4, it holds that either f ( y ) Z ( R ) for all y R or R is commutative. By using Lemma 2, it follows that R is commutative. This is a contradiction which completes the proof.

Let f be a nonzero anti-homomorphism of R. It implies that either R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. From the hypothesis, it gets that f ( [ x , y ] ) Z ( R ) for all x , y R . Since f is an anti-homomorphism, it is obtained that for all x , y R

f ( [ x , y ] ) = f ( x y y x ) = f ( y ) f ( x ) f ( x ) f ( y ) = [ f ( x ) , f ( y ) ]

i.e., for all x , y R

[ f ( x ) , f ( y ) ] Z ( R ) .

After here, the proof is done by the similar way in the first case and same result is obtained.

Theorem 6 If f is a nonzero homomorphism (or an anti-homomorphism) and f ( ( x , y ) ) = 0 for all x , y R then R is commutative.

Proof. Let f be a homomorphism of R. It holds that R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case, f is a right *-centralizer on R. So, it gets that for all x , y R

0 = f ( ( x , y ) ) = f ( x y + y x ) = f ( x y ) + f ( y x ) = f ( x ) f ( y ) + f ( y ) f ( x ) .

It means that for all x , y R

f ( x ) f ( y ) + f ( y ) f ( x ) = 0.

Replacing x by x z where z R in the above equation and using that f is a right * the last equation, it is obtained that

0 = f ( x z ) f ( y ) + f ( y ) f ( x z ) = x f ( z ) f ( y ) + f ( y ) x f ( z ) .

Using that f ( x ) f ( y ) = f ( y ) f ( x ) for all x , y R in the last equation

0 = x f ( z ) f ( y ) + f ( y ) x f ( z ) = x f ( y ) f ( z ) + f ( y ) x f ( z ) = [ f ( y ) , x ] f ( z )

i.e. for all x , y , z R

[ f ( y ) , x ] f ( z ) = 0.

Replacing x by xr, it follows that [ f ( y ) , x ] R f ( z ) = ( 0 ) for all x , y , z R . Using primeness of R, it holds that either [ f ( y ) , x ] = 0 or f ( z ) = 0 for all x , y , z R . Since f is nonzero, it implies that f ( R ) Z ( R ) . Using Lemma 2, it yields that R is commutative. This is a contradiction which completes the proof.

Let f be an anti-homomorphism of R. It holds that R is commutative or f is a right *-centralizer on R from Theorem 1. Assume that R is noncommutative. In this case f is a right *-centralizer on R. Using hypothesis, it gets that for all x , y R

0 = f ( ( x , y ) ) = f ( x y + y x ) = f ( x y ) + f ( y x ) = f ( y ) f ( x ) + f ( x ) f ( y )

i.e., for all x , y R

f ( y ) f ( x ) + f ( x ) f ( y ) = 0.

After here, the proof is done by the similar way in the first case and same result is obtained.

Now, g : R R is a generalized *-α-derivation associated with a *-α-derivation t on R.

Theorem 7 Let R be a *-prime ring where * be an involution, α be a homomorphism of R and g : R R be a generalized *-α-derivation associated with a *-α-derivation t on R. If g is nonzero then R is commutative.

Proof. Since g is a generalized *-α-derivation associated with a *-α-derivation t on R, it holds that g ( x y ) = g ( x ) y + α ( x ) t ( y ) for all x , y R . So it yields that for all x , y , z R

g ( x y z ) = g ( ( x y ) z ) = g ( x y ) z + α ( x y ) t ( z ) = ( g ( x ) y + α ( x ) t ( y ) ) z + α ( x ) α ( y ) t ( z ) = g ( x ) y z + α ( x ) t ( y ) z + α ( x ) α ( y ) t ( z )

that is, it holds that for all x , y , z R

g ( x y z ) = g ( x ) y z + α ( x ) t ( y ) z + α ( x ) α ( y ) t ( z ) . (5)

On the other hand, it implies that for all x , y , z R

g ( x y z ) = g ( x ( y z ) ) = g ( x ) ( y z ) + α ( x ) t ( y z ) = g ( x ) z y + α ( x ) ( t ( y ) z + α ( y ) t ( z ) ) = g ( x ) z y + α ( x ) t ( y ) z + α ( x ) α ( y ) t ( z )

so, it gets that for all x , y , z R

g ( x y z ) = g ( x ) z y + α ( x ) t ( y ) z + α ( x ) α ( y ) t ( z ) . (6)

Now, combining the Equations (5) and (6), it holds that for all x , y , z R

g ( x ) y z + α ( x ) t ( y ) z + α ( x ) α ( y ) t ( z ) = g ( x ) z y + α ( x ) t ( y ) z + α ( x ) α ( y ) t ( z )

which follows that

g ( x ) [ y , z ] = 0

for all x , y , z R . Replacing y by y and z by z , it holds that for all x , y , z R

g ( x ) [ y , z ] = 0.

Replacing y by ry where r R in the last equation, it yields that for all x , y , z , r R

0 = g ( x ) [ r y , z ] = g ( x ) r [ y , z ] + g ( x ) [ r , z ] y .

Using g ( x ) [ y , z ] = 0 for all x , y , z R in above equation, it is obtained that for all x , y , z , r R

g ( x ) r [ y , z ] = 0 (7)

i.e., for all x , y , z R

g ( x ) R [ y , z ] = ( 0 ) . (8)

Replacing y by y and z by z , it follows that for all x , y , z R

g ( x ) R ( [ y , z ] ) = ( 0 ) . (9)

Now, combining the Equations (8) and (9),

g ( x ) R [ y , z ] = g ( x ) R ( [ y , z ] ) = ( 0 )

is obtained for all x , y , z R . Using *-primeness of R, it follows that g ( x ) = 0 or [ y , z ] = 0 for all x , y , z R . Since g is nonzero, R is commutative.

Theorem 8 Let R be a semiprime *-ring where * be an involution, α be an homomorphism of R and g : R R be a nonzero generalized *-α-derivation associated with a *-α-derivation t on R then g ( R ) Z ( R ) .

Proof. Equation (7) multiplies by s from left, it gets that for all x , y , z , r , s R

s g ( x ) r [ y , z ] = 0. (10)

Replacing r by sr in the Equation (7), it holds that for all x , y , z , r , s R

g ( x ) s r [ y , z ] = 0. (11)

Now, combining the Equation (10) and (11),

s g ( x ) r [ y , z ] = g ( x ) s r [ y , z ]

is obtained for all x , y , z , r , s R . It follows that for all x , y , z , r , s R

[ s , g ( x ) ] r [ y , z ] = 0.

This implies that

[ s , g ( x ) ] R [ y , z ] = ( 0 )

for all x , y , z , s R . Replacing s by y and z by g ( x ) in the last equation, it yields that

[ y , g ( x ) ] R [ y , g ( x ) ] = ( 0 )

for all x , y R . Using semiprimeness of R, it is implied that for all x , y R

[ y , g ( x ) ] = 0.

That is,

g ( R ) Z ( R )

which completes the proof.

Cite this paper
Balcı, A. , Aydin, N. and Türkmen, S. (2018) The Commutativity of a *-Ring with Generalized Left *-α-Derivation. Advances in Pure Mathematics, 8, 168-177. doi: 10.4236/apm.2018.82009.
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