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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.

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 [

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 [

In [

The aim of this paper is to extend the results which proved for generalized left *-derivation of R in [

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

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.

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