The Commutativity of a *-Ring with Generalized Left *-α-Derivation

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.


Introduction
Let R be an associative ring with center ( ) A *-derivation on a *-ring was defined by Bresar and Vukman in [2] as follows: An additive mapping 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 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 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 ( ) ( ) ( ) ( ) In [4], Bell and Kappe proved that if : is a derivation holds as a homomorphism or an anti-homomorphism on a nonzero right ideal of R which is a prime ring, then 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 → is a generalized left derivation associated with a Jordan left derivation : R R δ → 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 ( ) → 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.

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 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 , , On the other hand, it holds that for all , , Combining Equation ( 1) and ( 2), it is obtained that for all , , This yields that for all , , Replacing y by yr where r R ∈ in the last equation, it implies that ∈ .Since α is surjective and R is prime, it follows that for all Replacing x by xy where y R ∈ in the last equation, it holds that for all , , Using Equation (3) in the last equation, it implies that for all , , Since α is surjective, it holds that for all , , Replacing z by x * in the last equation, it follows that for all , Since α is a surjective, it holds that ( ) Assume that B R = .This means that ( ) 0 d x = 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 Replacing y by xy in the last equation and using that f is an anti-homomorphism, it follows that for all , Replacing y by zy where z R ∈ in the last equation, it holds that for all , , 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 ( ) 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 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 which follows that for all , , x r = or ( ) 0 f y = for all , , x y r R ∈ .Since f is nonzero, it means that R is commutative.This is a contradiction which completes the proof.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 [ ] ( ) i.e., for all , 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 follows that for all , , Replacing x by xr where r R ∈ and using the last equation, it holds that ( ) ( ) x y z r R ∈ .This implies that for all , , Using the primeness of R, it is obtained that either ( ) 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 ([ , ]) 0 f x y = for all , x y R ∈ .Since f is an anti-homomorphism, it holds that for all , i.e., for all , 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 ( ), 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 ∈ ( ) i.e., for all , 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 homo-Advances in Pure Mathematics i.e., for all , x y r R ∈ .Replacing x by x z * where z R ∈ in the last equation, it holds that for all , , , which implies that for all , , , Replacing x by ( ) f y and r by ( ) The last equation multiplies by r from right and using that i.e., for all , , , x y z r R ∈ .
( ) ( ) ( ) ( ) ( ) Using primeness of R, it is implied that for all , From Theorem 4, it holds that either ( ) ( ) 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, 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 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 all , where z R ∈ in the above equation and using that f is a right * the last equation, it is obtained that i.e. for all , , Replacing x by xr, it follows that ( ) ( ) ( ) x y z R ∈ .
Using primeness of R, it holds that either ( ), . 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 f y f x f x f y + = 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 ( ) ( ) ( ) ( ) anti-homomorphism of R. A homomorphism of R is called an epimorphism if it is surjective.A ring R is called a prime if private, if b a = , it implies that R is a semiprime ring.An additive mapping :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 *-derivation and generalized left *-derivation were given in abu : Let R be a *-ring.An additive mapping : ∈ .It is clear that a generalized left *-derivation associated with zero mapping is a right *-centralizer on a *-ring.

Theorem 3
If f is a nonzero homomorphism (or an anti-homomorphism) and

,
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.
* in the last equation, it gets that [ ]