Advances in Pure Mathematics
Vol.08 No.02(2018), Article ID:82610,10 pages
10.4236/apm.2018.82009
The Commutativity of a *-Ring with Generalized Left *-α-Derivation
Ahmet Oğuz Balcı1, Neşet Aydin1, Selin Türkmen2
1Department of Mathematics, Faculty of Arts and Sciences, Çanakkale Onsekiz Mart University, Çanakkale, Turkey
2Lapseki Vocational School, Çanakkale Onsekiz Mart University, Çanakkale, Turkey
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/
Received: December 21, 2017; Accepted: February 23, 2018; Published: February 26, 2018
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 . where is denoted by and where is denoted by which holds some properties: and . An additive mapping α which holds for all is called a homomorphism of R. An additive mapping β which holds for all 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 implies that either or for fixed . In private, if , it implies that R is a semiprime ring. An additive mapping which holds and for all 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 implies that either or for fixed .
Notations of left *-derivation and generalized left *-derivation were given in : Let R be a *-ring. An additive mapping is called a left *-derivation if holds for all . An additive mapping is called a generalized left *-derivation if there exists a left *-derivation d such that holds for all . An additive mapping is called a right *-centralizer if for all . 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 is said to be a *-derivation if for all .
A generalized *-derivation on a *-ring was defined by Shakir Ali in Shakir: An additive mapping is said to be a generalized *-derivation if there exists a *-derivation such that for all .
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 such that for all 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 for all . 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 for all 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 holds for all .
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 . In [5] , Rehman proved that if is a nonzero generalized derivation with a nonzero derivation 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 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 for all and . In [1] , it is proved that if 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 is an involution, α is an epimorphism on R and 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
That is, it holds for all
(1)
On the other hand, it holds that for all
So, it means that for all
(2)
Combining Equation (1) and (2), it is obtained that for all
This yields that for all
Replacing y by yr where in the last equation, it implies that
for all . Since α is surjective and R is prime, it follows that for all
(3)
Replacing x by xy where 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 in the last equation, it follows that for all
Since α is a surjective, it holds that for all . Replacing y by yz where in the last equation, it gets for all . So, it implies that for all
Since R is prime, it follows that or for all . Let 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.
Assume that . This means that for all . Replacing x by in the last equation, it gets that for all . Therefore, R is commutative.
Assume that . This means that for all . 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
for all . It means that for all
Replacing y by xy in the last equation and using that f is an anti-homomorphism, it follows that for all
which implies that for all
(4)
Replacing y by zy where in the last equation, it holds that for all
Using Equation (4) in the above equation, it gets for all . Since is surjective, it holds that for all . That is, for all
Since R is prime, it implies that or for all . Let and . 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 . This means that for all . Since α is surjective, it holds that for all . It follows that R is commutative.
Assume that . 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 is in the center of R, it holds that for all . Using that f is a right *-centralizer and , it yields that for all
which follows that for all
Since is in the center of R, it is obtained that for all
Using primeness of R, it is implied that either or for all . 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 for all 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 for all . Since f is a homomorphism, it holds that for all
i.e., for all
Replacing x by in the last equation, using that f is a right *-centralizer on R and using the last equation, it holds that for . So, it follows that for all
Replacing x by xr where and using the last equation, it holds that for all . This implies that for all
Using the primeness of R, it is obtained that either or for all . Since f is nonzero, it follows that . 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 for all . 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), and for all then 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
i.e., for all
Replacing x by xr where , it holds that for all . This implies that for all . Using the primeness of R, it implies that or for all . Since f is nonzero, it follows that . That is, it is obtained that either or R is commutative.
Theorem 5 If f is a nonzero homomorphism (or an anti-homomorphism) and 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 homomorphism and for all , it holds that for all
i.e., for all
It means that for all . Replacing x by where in the last equation, it holds that for all
which implies that for all
Replacing x by and r by , it is obtained that for all
The last equation multiplies by r from right and using that for all , it follows that for all
i.e., for all .
Using primeness of R, it is implied that for all
From Theorem 4, it holds that either for all 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 for all . Since f is an anti-homomorphism, it is obtained that for all
i.e., for all
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 for all 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
It means that for all
Replacing x by where in the above equation and using that f is a right * the last equation, it is obtained that
Using that for all in the last equation
i.e. for all
Replacing x by xr, it follows that for all . Using primeness of R, it holds that either or for all . Since f is nonzero, it implies that . 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
i.e., for all
After here, the proof is done by the similar way in the first case and same result is obtained.
Now, 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 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 for all . So it yields that for all
that is, it holds that for all
(5)
On the other hand, it implies that for all
so, it gets that for all
(6)
Now, combining the Equations (5) and (6), it holds that for all
which follows that
for all . Replacing y by and z by , it holds that for all
Replacing y by ry where in the last equation, it yields that for all
Using for all in above equation, it is obtained that for all
(7)
i.e., for all
(8)
Replacing y by and z by , it follows that for all
(9)
Now, combining the Equations (8) and (9),
is obtained for all . Using *-primeness of R, it follows that or for all . 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 be a nonzero generalized *-α-derivation associated with a *-α-derivation t on R then .
Proof. Equation (7) multiplies by s from left, it gets that for all
(10)
Replacing r by sr in the Equation (7), it holds that for all
(11)
Now, combining the Equation (10) and (11),
is obtained for all . It follows that for all
This implies that
for all . Replacing s by y and z by in the last equation, it yields that
for all . Using semiprimeness of R, it is implied that for all
That is,
which completes the proof.
Cite this paper
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
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 Mathematics, 1, 32-37.
- 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