The Commutativity of a *-Ring with Generalized Left *-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.