Advances in Pure Mathematics, 2018, 8, 168-177 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 The Commutativity of a *-Ring with Generalized Left *-α-Derivation Ahmet Oğuz Balcı1, Neşet Aydin1, Selin Türkmen2 1 Department of Mathematics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart University, Çanakkale, Turkey Lapseki Vocational School, Çanakkale Onsekiz Mart University, Çanakkale, Turkey 2 How to cite this paper: Balcı, 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 Received: December 21, 2017 Accepted: February 23, 2018 Published: February 26, 2018 Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access 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. Keywords *-Ring, Prime *-Ring, Generalized Left *-α-Derivation, Generalized *-α-Derivation 1. Introduction Let R be an associative ring with center Z ( R ) . xy + yx where x, y ∈ R is denoted by ( x, y ) and xy − yx where x, y ∈ R is denoted by [ x, y ] which xy, z ] x [ y, z ] + [ x, z ] y and [= x, yz ] holds some properties: [= [ x, y ] z + y [ x, z ] . An additive mapping α which holds α ( xy ) = α ( x ) α ( y ) for all x, y ∈ R is called a homomorphism of R. An additive mapping β which holds β ( xy ) = 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 aRb = ( 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 ( xy ) = 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 = aRb aRb = (0) DOI: 10.4236/apm.2018.82009 Feb. 26, 2018 168 Advances in Pure Mathematics A. O. Balcı et al. implies that either a = 0 or b = 0 for fixed a, b ∈ R . Notations of left *-derivation and generalized left *-derivation were given in abu : Let R be a *-ring. An additive mapping d : R → R is called a left d ( xy ) x∗ d ( y ) + yd ( x ) holds for all x, y ∈ R . An additive *-derivation if = mapping F : R → R is called a generalized left *-derivation if there exists a left F ( xy ) x∗ F ( y ) + yd ( x ) holds for all x, y ∈ R . An *-derivation d such that = additive mapping T : R → R is called a right *-centralizer if T ( xy ) = 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 ( xy ) d ( x ) y ∗ + xd ( 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 exF ( xy ) F ( x ) y ∗ + xd ( y ) for all ists a *-derivation d : R → R such that = 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 ( xy ) 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 f ( xy ) x∗ f ( y ) + α ( y ) d ( x ) for all x, y ∈ R . left *-α-derivation d such that = 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 homomort ( xy ) t ( x ) y ∗ + α ( x ) t ( y ) for phism of R. An additive mapping t which holds = 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 ( xy ) 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 ) = xq for all x ∈ R and q ∈ Ql ( RC ) . In [1], it is proved that if F : R → R is a generalized left *-derivation associated with a left DOI: 10.4236/apm.2018.82009 169 Advances in Pure Mathematics A. O. Balcı et al. *-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 ( xyz ) f (= x ( yz ) ) x∗ f ( yz ) + α ( yz ) d ( x ) = = x∗ f ( y ) f ( z ) + α ( y ) α ( z ) d ( x ) . That is, it holds for all x, y, z ∈ R = f ( xyz ) 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 ( xyz ) f= xy ) f ( z ) x∗ f ( y ) f ( z ) + α ( y ) d ( x ) f ( z ) . ( ( xy ) z ) f (= So, it means that for all x, y, z ∈ R = f ( xyz ) 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 DOI: 10.4236/apm.2018.82009 170 Advances in Pure Mathematics A. O. Balcı et al. 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 ∗ 0. α ( z ) , x  d ( y ) + α ( z ) , α ( y )  d ( x ) = Since α is surjective, it holds that for all x, y, z ∈ R ∗ 0.  z, x  d ( y ) +  z, α ( y )  d ( x ) = 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  zd ( x ) = 0 for all x, y, z ∈ R . So, it implies that for all x, y ∈ R  x∗ , y  Rd ( 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 0} . Both A and B are and B = {x ∈ R | d ( x ) = 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 R is commutative. [ x, y ] = 0 for all x, y ∈ R . Therefore, 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 ( xy ) 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 DOI: 10.4236/apm.2018.82009 171 Advances in Pure Mathematics A. O. Balcı et al. that  z, α ( x )  yd ( x ) = 0 for all x, y, z ∈ R . That is, for all x, z ∈ R  z , α ( x )  Rd ( 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 = 0} . Both K and {x ∈ R | d ( x ) = 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 com- mutative. 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  = xf ( 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 ] Rf ( 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 contra- diction 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 ( xy − yx )= 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 DOI: 10.4236/apm.2018.82009 172 Advances in Pure Mathematics A. O. Balcı et al. ( ) 0  f= x∗ z , f ( y )  = xf ( 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 )  rf ( z ) = 0 for all x, y, z, r ∈ R . This implies that for all x, y, z ∈ R  x, f ( y )  Rf ( 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 ( xy − yx ) = 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  =  xf ( 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 ] rf ( y ) = 0 for all x, y, r ∈ R . This implies that [ x, a ] Rf ( y ) = ( 0 ) for all x, y ∈ R . Using the primeness of R, it implies that is nonzero, it follows that [ x, a ] = 0 or f ( y ) = 0 for all x, y ∈ R . Since f 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 homoDOI: 10.4236/apm.2018.82009 173 Advances in Pure Mathematics A. O. Balcı et al. ([ x, y ]) ∈ Z ( R ) for all x, y ∈ R , it holds that for all f ([ x, y ])= f ( xy − yx )= f ( xy ) − f ( yx ) morphism and f x, y ∈ R = 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 ]   xf ( 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 0. [ x, r ]  f ( z ) , f ( y ) +   x, f ( y ) , r  f ( z ) +  x, f ( y )  f ( z ) , r  = 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 f ( xy − yx ) = f ( y) f ( x) − f ( x) f ( y) = −  f ( x ) , f ( y )  ( [ x, 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 DOI: 10.4236/apm.2018.82009 ( ( x, y ) ) = 0 for all x, y ∈ R then R is commutative. 174 Advances in Pure Mathematics A. O. Balcı et al. 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 ( xy + yx )= f ( xy ) + f ( yx )= 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 =xf ( z ) f ( y ) + f ( y ) xf ( z ) . Using that f ( x ) f ( y ) = − f ( y ) f ( x ) for all x, y ∈ R in the last equation 0= xf ( z ) f ( y ) + f ( y ) xf ( z ) = − xf ( y ) f ( z ) + f ( y ) xf ( 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  Rf ( 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 ( xy + yx )= f ( xy ) + f ( yx )= 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 ( xy ) g ( x ) y ∗ + α ( x ) t ( y ) for all x, y ∈ R . So it yields that for all x, y, z ∈ R = g ( xyz ) g (= ( xy ) z ) g ( xy ) z∗ + α ( xy ) 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 ) DOI: 10.4236/apm.2018.82009 175 Advances in Pure Mathematics A. O. Balcı et al. that is, it holds that for all x, y, z ∈ R g ( xyz ) = 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 ( xyz ) g= = ( x ( yz ) ) g ( x )( yz ) + α ( x ) t ( yz ) ∗ ( 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 ( xyz ) = 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 ) [ ry, 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) g ( x ) R [ y, z ] = ( 0 ) . (8) i.e., for all x, y, z ∈ R 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 sg ( x ) r [ y, z ] = 0. DOI: 10.4236/apm.2018.82009 176 (10) Advances in Pure Mathematics A. O. Balcı et al. Replacing r by sr in the Equation (7), it holds that for all x, y, z , r , s ∈ R g ( x ) sr [ y, z ] = 0. (11) Now, combining the Equation (10) and (11), sg ( x ) r [ y, z ] = g ( x ) sr [ 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. References [1] Rehman, N., Ansari, A.Z. and Haetinger, C. (2013) A Note on Homomorphisms and Anti-Homomorphisms on *-Ring. Thai Journal of Mathematics, 11, 741-750. [2] Bresar, M. and Vukman, J. (1989) On Some Additive Mappings in Rings with Involution. Aequationes Mathematicae, 38, 178-185. https://doi.org/10.1007/BF01840003 [3] Ali, S. (2012) On Generalized *-Derivations in *-Rings. Palestine Journal of Mathe- matics, 1, 32-37. DOI: 10.4236/apm.2018.82009 [4] Bell, H.E. and Kappe, L.C. (1989) Ring in Which Derivations Satisfy Certain Algebraic Conditions. Acta Mathematica Hungarica, 53, 339-346. https://doi.org/10.1007/BF01953371 [5] Rehman, N. (2004) On Generalized Derivations as Homomorphisms and Anti-Homomorphisms. Glasnik Matematicki, 39, 27-30. https://doi.org/10.3336/gm.39.1.03 [6] Dhara, B. (2012) Generalized Derivations Acting as a Homomorphism or Anti-Homomorphism in Semiprime Rings. Beiträge zur Algebra und Geometrie, 53, 203-209. https://doi.org/10.1007/s13366-011-0051-9 [7] Ali, S. (2011) On Generalized Left Derivations in Rings and Banach Algebras. Aequationes Mathematicae, 81, 209-226. https://doi.org/10.1007/s00010-011-0070-5 177 Advances in Pure Mathematics