Let R be a semiprime ring, I a nonzero ideal of R, and σ, τ two epimorphisms of R. An additive mapping F:R→R is generalized (σ,τ)-derivation on R if there exists a (σ,τ)-derivation d:R→R such that F(xy)=F(x)σ(y)+τ(x)d(y) holds for all x,y∈R. In this paper, it is shown that if τ(I)d(I)≠0, then R contains a nonzero central ideal of R, if one of the following holds: (i) F[x,y]=±(x∘y)σ,τ; (ii) F(x∘y)=±[x,y]σ,τ; (iii) F[x,y]=±[F(x),y]σ,τ; (iv) F(x∘y)=±(F(x)∘y)σ,τ; (v) F[x,y]=±[σ(y),G(x)] for all x,y∈I.

1. Introduction

Throughout the present paper, R always denotes an associative semiprime ring with center Z(R). For any x,y∈R, the commutator and anticommutator of x and y are denoted by [x,y] and x∘y and are defined by xy-yx and xy+yx, respectively. Recall that a ring R is said to be prime, if for a,b∈R, aRb=0 implies either a=0 or b=0 and is said to be semiprime if for a∈R, aRa=0 implies a=0. An additive mapping d:R→R is said to be derivation if d(xy)=d(x)y+xd(y) holds for all x,y∈R. The notion of derivation is extended to generalized derivation. The generalized derivation means an additive mapping F:R→R associated with a derivation d:R→R such that F(xy)=F(x)y+xd(y) holds for all x,y∈R. Then every derivation is a generalized derivation, but the converse is not true in general.

A number of authors have studied the commutativity theorems in prime and semiprime rings admitting derivation and generalized derivation (see e.g., [1–8]; where further references can be found).

Let α and β be two endomorphisms of R. For any x,y∈R, set [x,y]α,β=xα(y)-β(y)x and (x∘y)α,β=xα(y)+β(y)x. An additive mapping d:R→R is called a (α,β)-derivation if d(xy)=d(x)α(y)+β(x)d(y) holds for all x,y∈R. By this definition, every (1,1)-derivation is a derivation, where 1 means the identity map of R. In the same manner the concept of generalized derivation is also extended to generalized (α,β)-derivation as follows. An additive map F:R→R is called a generalized (α,β)-derivation if there exists a (α,β)-derivation d:R→R such that F(xy)=F(x)α(y)+β(x)d(y) holds for all x,y∈R. Of course every generalized (1,1)-derivation is a generalized derivation of R, where 1 denotes the identity map of R.

There is also ongoing interest to study the commutativity in prime and semiprime rings with (α,β)-derivations or generalized (α,β)-derivations (see [9–17]).

The present paper is motivated by the results of [17]. In [17], Rehman et al. have discussed the commutativity of a prime ring on generalized (α,β)-derivation, where α and β are automorphisms of R. More precisely, they studied the following situations: (i) F[x,y]=±(x∘y)σ,τ; (ii) F(x∘y)=±[x,y]σ,τ; (iii) F[x,y]=±[F(x),y]σ,τ; (iv) F(x∘y)=±(F(x)∘y)σ,τ; (v) F[x,y]=±[σ(y),G(x)] for all x,y∈I, where I is a nonzero ideal of R.

The main objective of the present paper is to extend above results for generalized (α,β)-derivations in semiprime ring R, where α and β are considered as epimorphisms of R.

To prove our theorems, we will frequently use the following basic identities:
(1.1)[xy,z]α,β=x[y,z]α,β+[x,β(z)]y=x[y,α(z)]+[x,z]α,βy,[x,yz]α,β=β(y)[x,z]α,β+[x,y]α,βα(z),(x∘(yz))α,β=(x∘y)α,βα(z)-β(y)[x,z]α,β=β(y)(x∘z)α,β+[x,y]α,βα(z),((xy)∘z)α,β=x(y∘z)α,β-[x,β(z)]y=(x∘z)α,βy+x[y,α(z)].

2. Main ResultsTheorem 2.1.

Let R be a semiprime ring, I a nonzero ideal of R, σ and τ two epimorphisms of R and F a generalized (σ,τ)-derivation associated with a (σ,τ)-derivation d of R such that τ(I)d(I)≠0. If F([x,y])=±(x∘y)σ,τ for all x,y∈I, then R contains a nonzero central ideal.

Proof.

First we consider the case
(2.1)F([x,y])=(x∘y)σ,τ
for all x,y∈I. Replacing y by yx in (2.1) we get
(2.2)F([x,y])σ(x)+τ([x,y])d(x)=(x∘y)σ,τσ(x)-τ(y)[x,x]σ,τ.
Using (2.1), it reduces to
(2.3)τ([x,y])d(x)=-τ(y)[x,x]σ,τ
for all x,y∈I. Again replacing y by ry in (2.3), we get
(2.4){τ(r)τ([x,y])+τ([x,r])τ(y)}d(x)=-τ(r)τ(y)[x,x]σ,τ
for all x,y∈R and r∈R. Left multiplying (2.3) by τ(r) and then subtracting from (2.4) we have
(2.5)τ([x,r])τ(y)d(x)=0
for all x,y∈I and r∈R. Replacing y with sy, s∈R, we get τ([x,r])τ(s)τ(y)d(x)=0 for all x,y∈I and r,s∈R. Since τ is an epimorphism of R, we can write
(2.6)[R,τ(x)]Rτ(I)d(x)=0
for all x∈I.

Since R is semiprime, it must contain a family Ω={Pα∣α∈Λ} of prime ideals such that ∩Pα={0}. If P is a typical member of Ω and x∈I, it follows that
(2.7)[R,τ(x)]⊆Porτ(I)d(x)⊆P.

Construct two additive subgroups T1={x∈I∣[R,τ(x)]⊆P} and T2={x∈I∣τ(I)d(x)⊆P}. Then T1⋃T2=I. Since a group cannot be a union of two its proper subgroups, either T1=I or T2=I, that is, either [τ(I),R]⊆P or τ(I)d(I)⊆P. Thus both cases together yield [R,τ(I)]τ(I)d(I)⊆P for any P∈Ω. Therefore, [R,τ(I)]τ(I)d(I)⊆⋂α∈ΛPα=0, that is, [R,τ(I)]τ(I)d(I)=0. Thus
(2.8)0=[R,τ(RIR)]τ(RI)d(I)=[R,Rτ(I)R]Rτ(I)d(I)
and so 0=[R,Rτ(I)d(I)R]Rτ(I)d(I)R. This implies 0=[R,J]RJ, where J=Rτ(I)d(I)R is a nonzero ideal of R, since τ(I)d(I)≠0. Then 0=[R,J]R[R,J]. Since R is semiprime, it follows that 0=[R,J], that is, J⊆Z(R).

Similarly, we can obtain the same conclusion when F([x,y])=-(x∘y)σ,τ for all x,y∈I.

Theorem 2.2.

Let R be a semiprime ring, I a nonzero ideal of R, σ and τ two epimorphisms of R and F a generalized (σ,τ)-derivation associated with a (σ,τ)-derivation d of R such that τ(I)d(I)≠0. If F(x∘y)=±[x,y]σ,τ for all x,y∈I, then R contains a nonzero central ideal.

Proof.

We begin with the case
(2.9)F(x∘y)=[x,y]σ,τ
for all x,y∈I. Replacing y by yx in (2.9) we get
(2.10)F(x∘y)σ(x)+τ(x∘y)d(x)=τ(y)[x,x]σ,τ+[x,y]σ,τσ(x).
Right multiplying (2.9) by σ(x) and then subtracting from (2.10) we get
(2.11)τ(x∘y)d(x)=τ(y)[x,x]σ,τ
for all x,y∈I.

Now replacing y by ry in (2.11), we obtain
(2.12)τ(r)τ(x∘y)d(x)+τ([x,r])τ(y)d(x)=τ(r)τ(y)[x,x]σ,τ
for all x,y∈I and for all r∈R. Left multiplying (2.11) by τ(r) and then subtracting from (2.12), we get
(2.13)τ([x,r])τ(y)d(x)=0
for all x,y∈I and for all r∈R. This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar results hold in case F(x∘y)=-[x,y]σ,τ for all x,y∈I.

Theorem 2.3.

Let R be a semiprime ring, I a nonzero ideal of R, σ and τ two epimorphisms of R and F a generalized (σ,τ)-derivation associated with a (σ,τ)-derivation d of R such that τ(I)d(I)≠0. If F[x,y]=±[F(x),y]σ,τ for all x,y∈R, then R contains a nonzero central ideal.

Proof.

We assume first that F[x,y]=[F(x),y]σ,τ for all x,y∈I. This implies
(2.14)F(x)σ(y)+τ(x)d(y)-F(y)σ(x)-τ(y)d(x)=[F(x),y]σ,τ.
Replacing y by yx in (2.14) we have
(2.15)F(x)σ(y)σ(x)+τ(x){d(y)σ(x)+τ(y)d(x)}-F(y)σ(x)2-τ(y)d(x)σ(x)-τ(y)τ(x)d(x)=τ(y)[F(x),x]σ,τ+[F(x),y]σ,τσ(x).
Right multiplying (2.14) by σ(x) and then subtracting from (2.15), we get
(2.16)τ(x)τ(y)d(x)-τ(y)τ(x)d(x)=τ(y)[F(x),x]σ,τ.
Now replacing y by ry, where r∈R, in (2.16), we obtain
(2.17)τ(x)τ(r)τ(y)d(x)-τ(r)τ(y)τ(x)d(x)=τ(r)τ(y)[F(x),x]σ,τ.
Left multiplying (2.16) by τ(r) and then subtracting from (2.17), we get that
(2.18)[τ(x),τ(r)]τ(y)d(x)=0,
that is,
(2.19)τ([x,r])τ(y)d(x)=0
for all x,y∈I and for all r∈R. This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar results hold in case F[x,y]=-[F(x),y]σ,τ for all x,y∈I.

Theorem 2.4.

Let R be a semiprime ring, I a nonzero ideal of R, σ and τ two epimorphisms of R and F a generalized (σ,τ)-derivation associated with a (σ,τ)-derivation d of R such that τ(I)d(I)≠0. If F(x∘y)=±(F(x)∘y)σ,τ for all x,y∈I, then R contains a nonzero central ideal.

Proof.

By our assumption first consider F(x∘y)=(F(x)∘y)σ,τ for all x,y∈I. This gives
(2.20)F(x)σ(y)+τ(x)d(y)+F(y)σ(x)+τ(y)d(x)=(F(x)∘y)σ,τ.
Replacing y by yx in (2.20), we have
(2.21)F(x)σ(y)σ(x)+τ(x){d(y)σ(x)+τ(y)d(x)}+{F(y)σ(x)+τ(y)d(x)}σ(x)+τ(y)τ(x)d(x)=(F(x)∘y)σ,τσ(x)-τ(y)[F(x),x]σ,τ.
Right multiplying (2.20) by σ(x) and then subtracting from (2.21), we obtain that
(2.22)τ(x)τ(y)d(x)+τ(y)τ(x)d(x)=-τ(y)[F(x),x]σ,τ.
Now replacing y by ry, where r∈R, in (2.22) and by using (2.22), we obtain
(2.23)τ([x,r])τ(y)d(x)=0
for all x,y∈I and for all r∈R. This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar argument can be adapted in case F(x∘y)=-(F(x)∘y)σ,τ for all x,y∈I.

Theorem 2.5.

Let R be a semiprime ring, I a nonzero ideal of R, σ and τ two epimorphisms of R and F a generalized (σ,τ)-derivation associated with a nonzero (σ,τ)-derivation d of R such that τ(I)d(I)≠0. If F[x,y]=±[σ(y),G(x)] for all x,y∈I, then R contains a nonzero central ideal.

Proof.

We begin with the situation
(2.24)F[x,y]=[σ(y),G(x)]
for all x,y∈I. Replacing y by yx in (2.24), we get
(2.25)F([x,y])σ(x)+τ([x,y])d(x)=[σ(y)σ(x),G(x)].
Right multiplying (2.24) by σ(x) and then subtracting from (2.25), we obtain that
(2.26)τ([x,y])d(x)=σ(y)[σ(x),G(x)]
for all x,y∈I. Now replacing y by ry in (2.26), where r∈R, and by using (2.26), we obtain
(2.27)τ([x,r])τ(y)d(x)=0
for all x,y∈I and for all r∈R. This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

In case F[x,y]=-[α(y),G(x)] for all x,y∈I, the similar argument can be adapted to draw the same conclusion.

We know the fact that if a prime ring R contains a nonzero central ideal, then R must be commutative (see Lemma 2 in [18]). Hence the following corollary is straightforward.

Corollary 2.6.

Let R be a prime ring, σ and τ two epimorphisms of R and F a generalized (σ,τ)-derivation associated with a nonzero (σ,τ)-derivation d of R satisfying any one of the following conditions:

F([x,y])=(x∘y)σ,τ for all x,y∈R or F([x,y])=-(x∘y)σ,τ for all x,y∈R;

F(x∘y)=[x,y]σ,τ for all x,y∈R or F(x∘y)=-[x,y]σ,τ for all x,y∈R;

F[x,y]=[F(x),y]σ,τ for all x,y∈R or F[x,y]=-[F(x),y]σ,τ for all x,y∈R;

F(x∘y)=(F(x)∘y)σ,τ for all x,y∈R or F(x∘y)=-(F(x)∘y)σ,τ for all x,y∈R;

F[x,y]=[σ(y),G(x)] for all x,y∈R or F[x,y]=-[σ(y),G(x)] for all x,y∈R;

then R must be commutative.
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