Throughout this paper, R will be a ring with center Z(R). Let x, y∈ R. The commutator xy-yx will be denoted by [x,y] and the anti-commutator xy+yx will be represented by x∘y. Recall that an ideal P of R is prime if for all x,y∈R, xRy⊆P implies x∈P or y∈P. An additive mapping d:R→R is called a derivation if d(xy)=d(x)y+xd(y) holds for all x,y∈R. An additive mapping F:R→R is called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) for all x,y∈R, and d is called the associated derivation of F. During the past few decades, there has been an ongoing interest concerning the relationship between the commutativity of a ring and the existence of certain specific types of derivations of R.

An additive mapping T:R→R is said to be a left centralizer (resp. right centralizer) of R if T(xy)=T(x)y (resp. T(xy)=xT(y)) for all x,y∈R. An additive mapping T is called a centralizer in case T is a left and a right centralizer of R. In ring theory it is more common to work with module homomorphisms. Ring theorists would write that T:RR→RR is a homomorphism of a ring module R into itself. For a semi-prime ring R all such homomorphisms are of the form T(x) = qx for all x∈ R, where q is an element of Martindale left ring of quotients Q_r (see [5, Chapter 2]). If R has the identity element then T:R→R is a left centralizer if T is of the form T(x) = ax for all x ∈ R and some fixed element a ∈ R. Recently there has been a great interest in the study of the relationship between the commutativity of a ring and some specific additive mappings defined on the considered ring. In this direction, several authors have studied this problem by considering left (respectively right) centralizers in prime and semi-prime rings (see for example [1, 2, 6, 7], where further references can be found).

In the following definition, we have initiated the concept of I-centralizers in rings, where I is an ideal, and extended several known results.

Definition. Let I be an ideal of a ring R and f:R→R an additive mapping.

• (1) f is called an I-left centralizer if  f(xy)−f(x)y∈I  for all  x,y∈R.

• (2) f is called an I-right centralizer if  f(xy)−xf(y)∈I  for all  x,y∈R.

• (3) f is called an I-centralizer if and only if f is both an I-left centralizer and I-right centralizer.

Example.

• (1) The zero function ΘR is an I-centralizer on R.

• (2) The I_d and -I_d are I-left centralizers (resp. I-right centralizers) on R, where I_d denotes the identity function.

• (3) Consider the ring R=xy00000z0 | x,y,z∈ℤ. Let I be the nonzero ideal of R defined by I=αβ0000000 | α,β∈ℤ. It is easy to verify that the additive mapping T:R→R defined by:

  Txy00000z0=z00000000

  is an I-centralizer but T is not a centralizer.

The main goal of this work is to continue on this line of investigation and study the relationship between the structure of quotient rings R/P and the behavior of P-centralizers satisfying specific algebraic identities.

In the sequel, we shall make some use of the following well-known result.

Fact 1.1. Let R be a ring, I a nonzero ideal of R and P a prime ideal of R such that P⊈I. If aIb⊆P for all a,b∈ R, then a ∈ P or b ∈ P.

Fact 1.1. Let R be a semi-prime ring, I a nonzero ideal of R and a ∈ I such that aIa=0, then a=0.

In what follows, x¯ for x in R denotes x+P in R/P.

In [4][Theorem 2.3], Aydin proved that if R is a non-commutative prime ring, F a generalized derivation of R associated with a nonzero derivation d and a ∉ Z(R) such that F(x)a=aF(x) for all x ∈ I, then d(x)=λ[x,a], for all x∈ I, where I is an ideal of R.

Inspired by the above result, we here consider a more general algebraic identity involving two P-left centralizers by omitting the primeness assumption imposed on the ring R.

Theorem 2.1. Let R be a ring, I a nonzero ideal of R and P a prime ideal of R such that P⊈I. Suppose that T1 and T2 are two P-left centralizers on R, satisfying the condition T1(x)a−aT2(x)¯∈Z(R/P) for all x ∈ I, where a ∈ R, then one of the following assertions holds:

• (1) T1(R)⊆P and aT2(R)⊆P;

• (2) R/P is a commutative integral domain;

• (3) [a,R]⊂P.

Proof. By assumption, we have

Replacing x by xy in (2.1), we obtain

in such a way that

Substituting yr for y in (2.2), we get

That is

Putting T1(x)y instead of y in (2.3) and using it, one can see that

According to Fact 1.1, we obtain for each r∈ I, either[T1(x),r]T1(x)∈P or [r,a]∈P. Define A={r∈I/[T1(x),r]T1(x)∈P for all x∈I} and B={r∈I/[r,a]∈P}. Clearly, A and B are additive subgroups of I whose union is I. Hence by Brauer's trick, we have either A = I or B = I.

In the second case, namely [I,a]⊆P. Since RI⊆I, then [R,a]⊆P.

Now consider A=I, in this situation

Substituting sr for r in the above expression, we arrive at

Right multiplying the above equation by s and combining it with (2.4), it follows that

Applying Fact 1.2, we conclude that T1(x)¯∈Z(R/P) for all x∈ I. Writing xt for x in the last expression, where t∈ R, we arrive at t¯∈Z(R/P) or T1(x)¯=0¯. i.e., R/P is commutative or T1(R)⊆P and our hypothesis reduces to

which means that

Replacing x by xt in (2.5), on can see that

Right multiplying (2.5) by t and subtracting it from (2.6), we get

Substituting r by ru in (2.7) and employing it, we obtain

Once again invoking Fact 1.1, it follows from equation (2.8) that aT2(R)⊆P or [R,R]⊆P. Finally, we have either (T1(R)⊆P and aT2(R)⊆P) or [a,R]⊆P.

As an application of our Theorem, we get the following result.

Corollary 2.2. Let R be a non-commutative prime ring and I a nonzero ideal of R. Suppose that T₁ and T₂ are two left centralizers on R such that T1(x)a±aT2(x)∈Z(R) for all x∈ I, where a∈Z(R), then T₁=0 and aT₂=0.

In [3, Theorem 2.1], it is showed that if a prime ring R admits a nonzero left centralizer T, with T(x) ≠ x for all x in a nonzero ideal I of R, such that T([x,y])=[x,y] for all x,y ∈ I, then R must be commutative. The author in [8] with addition of 2-torsion freeness hypothesis, extended the preceding result to a Jordan ideal.

Motivated by the preceding results we investigate a more general context which allows us to generalize the above result in two ways. First of all, we will assume that T([x,y]) belong to center of R/P rather than T([x,y]) = 0. Secondly we will investigate the behavior of the more general expression T1(xy)−T2(yx)¯∈Z(R/P) involving two P-left centralizers instead of the expression T(xy)-T(yx)=0.

Theorem 2.3. Let R be a ring, I a nonzero ideal of R and P a prime ideal of R such that P⊈I. Suppose that T₁ and T₂ are two P-left centralizers on R, then the following assertions are equivalent:

• (1) T1(xy)−T2(yx)¯∈Z(R/P) for all x,y∈I;

• (2) T1(R)⊆P and  T2(R)⊆P or R/P is a commutative integral domain.

Proof. By given assumption, we have

Substituting yr for y in (2.12), and by expanding this equation, we get

Replacing y by yT2(y) in (2.10), we find that

In light of (2.10), Eq. (2.11) yields

Writing tx for x in (2.12), one can easily to see that

According to Fact 1.1, we obtain either R/P is an integral domain or [T2(y),r]T2(y)∈P for all r,y∈ I. Arguing as above, the last relation assures that T2(y)¯∈Z(R/P) for all y∈ I and our hypothesis becomes

Putting yu instead of y in (2.13), we get

By the primeness of P, we conclude that T1(R)⊆P or R/P is an integral domain. Now if T1(R)⊆P, then equation (2.9) yields T2(y)x¯∈Z(R/P) for all x,y∈ I. Commuting this expression with r, we find that T2(y)I[x,r]⊆P. Once again applying Fact 1.1, it follows that T2(R)⊆P or R/P is a commutative integral domain.

As an application of Theorem 2.3, the following corollary gives a generalization of some results in [3, 8].

Corollary 2.4. Let R be a prime ring and I a nonzero ideal of R. Suppose that T₁ and T₂ are nonzero two left centralizers on R, then the following assertions are equivalent:

• (1) T1(xy)±T2(yx)∈Z(R) for all x,y∈I;

• (2) R is a commutative integral domain.

Corollary 2.5. Let R be a prime ring and I a nonzero ideal of R. Suppose that T is a nonzero left centralizer on R, then the following assertions are equivalent:

• (1) T([x,y])∈Z(R) for all x,y∈I;

• (2) T(x∘y)∈Z(R) for all x,y∈I;

• (3) R
  is a commutative integral domain.

In [3, Theorems 3.1 and 3.3], it is proved that a prime ring R must be a commutative integral domain if it admits a non trivial left centralizer T such that T(xy)−xy∈Z(R) or T(xy)−yx∈Z(R) for all x,y in a nonzero ideal I of R. This result can be obtained as an immediate application of Corollary 2.5.

Corollary 2.6. Let R be a prime ring and I a nonzero ideal of R. Suppose that T is a non trivial left centralizer on R, then the following assertions are equivalent:

• (1) T(xy)±xy∈Z(R) for all x,y∈I;

• (2) T(xy)±yx∈Z(R) for all x,y∈I;

• (3) R is a commutative integral domain.

The following theorem exhibits a connection between the commutativity of R/P and range inclusion results of a pair of P-left centralizers.

Theorem 2.7. Let R be a ring, I a nonzero ideal of R and P a prime ideal of R such that P⊈I. If T₁ and T₂ are two P-left centralizers on R, then the following assertions are equivalent:

• (1) T1(x)T2(x)¯∈Z(R/P) for all x∈ I;

• (2) T1(R)⊆P or T2(R)⊆P or R/P is a commutative integral domain.

Proof. For non-trivial implications. Assume that

A Linearization of (2.14) gives

This means that

Substituting yT2(x) for y in (2.15) and combining it from the above expression, we get

Putting tr instead of r in (2.16), we obtain

In view of the primeness of P, we find that either T1(x)T2(y)+T1(y)T2(x)∈P for all x,y ∈ I or [T2(x),r]∈P for all r,x∈ I.

In the latter case, taking x=xs, it is obviously to see that

Writing xu for x and using Fact 1.1, we arrive at T2(R)⊆P or R/P is commutative.

Now consider the first case, i.e., T1(x)T2(y)+T1(y)T2(x)∈P for all x,y ∈ I. Replacing y by yw in this equation, it follows that T1(y)(T2(x)w−wT2(x))∈P for all w,x,y ∈ I. Thereby obtaining,

Therefore, either T1(R)⊆P or T2(x)w−wT2(x)∈P for all w,x ∈ I. In the last case, putting x=xy, we easily get T2(x)[w,y]∈P for all w,x,y ∈ I proving that T2(R)⊆P or R/P is an integral domain.

Corollary 2.8. Let R be a prime ring and I a nonzero ideal of R. If T₁ and T₂ are two nonzero left centralizers on R such that T1(x)T2(x)∈Z(R) for all x∈ I, then R is a commutative integral domain.

Theorem 2.9. Let R be a ring, I a nonzero ideal of R and P a prime ideal of R such that P⊈I. If T₁ and T₂ are two P-left centralizers on R, then the following assertions are equivalent:

• (1) [T1(x),T2(y)]¯∈Z(R/P) for all x,y∈I;

• (2) T1(x)∘T2(y)¯∈Z(R/P) for all x,y∈I;

• (3) T1(R)⊆P or T2(R)⊆P or R/P is a commutative integral domain.

Proof. Wee only need to prove (1)⇒ (3) and (2) ⇒ (3). (1) ⇒ (3) For all x,y∈I, we suppose that

This may be rewritten as

Analogously, replacing yt for y, where t∈ R in (2.19), and by appropriate expansion, get

Letting t=T1(x) in (2.20), one can see that

Keeping in mind that [T1(x),T2(y)]¯∈Z(R/P), we get

In light of the primeness of P, we find that either [T1(x),T2(y)]∈P or [T1(x),r]∈P for all x ∈ I. Consequently, I is a union of two additive subgroups I₁ and I₂, where

According to Brauer's trick, we are forced to conclude that either I=I₁ or I=I₂.

If I=I₁, i.e. [T1(x),T2(y)]∈P for all x,y∈I, then replacing y by ys, one obtains

Substituting yu for y in (2.22), we obviously get

So again an appeal to Fact 1.1, gives either T2(R)⊆P or [T1(x),s]∈P for all x,s∈ I.

Now if I=I₂, that is [T1(x),r]∈P for all x,r ∈ I, then putting xw instead of x, we obtain

Writing xw for x in (2.23), we get

Accordingly, it follows that T1(R)⊆P or R/P is a commutative integral domain.

(2) ⇒ (3) Can be proved by using the same steps as we did before.

Corollary 2.10. Let R be a prime ring and I a nonzero ideal of R. If T₁ and T₂ are two nonzero left centralizers on R, then the following assertions are equivalent:

• (1) [T1(x),T2(y)]∈Z(R) for all x,y∈I;

• (2) T1(x)∘T2(y)∈Z(R) for all x,y∈I;

• (3) R is a commutative integral domain.

Using similar arguments as above with necessary variation, we can prove the following theorem.

Theorem 2.11. Let R be a ring, I a nonzero ideal of R and P a prime ideal of R such that P⊈I. Suppose that T₁ and T₂ are two P-left centralizers on I of R, then the following assertions are equivalent:

• (1) T1(x)T2(y)−[x,y]¯∈Z(R/P) for all x,y∈I;

• (2) T1(x)T2(y)−x∘y¯∈Z(R/P) for all x,y∈I;

• (3) R/P is a commutative integral domain.

Let R be a prime ring. Letting P = (0) in the previous theorem, we deduce that, if T1(x)T2(y)−[x,y]∈Z(R) or T1(x)T2(y)−x∘y∈Z(R) for all x,y∈I, then R is commutative. The following corollary shows that the same conclusion remains satisfied for semi-prime rings.

Corollary 2.12. Let R be a semi-prime ring and I a nonzero ideal of R. Suppose that T₁ and T₂ are two left centralizers on R, then the following assertions are equivalent:

• (1) T1(x)T2(y)±[x,y]∈Z(R) for all x,y∈I;

• (2) T1(x)T2(y)±x∘y∈Z(R) for all x,y∈I;

• (3) R is commutative.

Proof. We have only to prove (1) ⇒ (3), while the implication (2) ⇒ (3) can be proved similarly. The ring R is semi-prime, then there exists a family P of prime ideals such that ∩ P∈PP=(0). Then we may suppose existence of a two left centralizers T₁ and T₂ satisfying T1(x)T2(y)±[x,y]∈Z(R) for all x, y∈ I. Thereby obtaining, [T1(x)T2(y)±[x,y],r]=0∈∩ P∈PP for all r,x,y ∈ I, therefore, Theorem 2.11 yields that for all P∈P, R/P is commutative which, because of ∩ P∈PP=(0), assures that R is commutative.