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### Article

Kyungpook Mathematical Journal 2023; 63(4): 551-560

Published online December 31, 2023 https://doi.org/10.5666/KMJ.2023.63.4.551

### Commutativity Criteria for a Factor Ring R/P Arising from P-Centralizers

Lahcen Oukhtite* and Karim Bouchannafa, My Abdallah Idrissi

Department of Mathematics, Faculty of Sciences and Technology, S. M. Ben Abdellah University, Fez, Morocco
e-mail : oukhtitel@hotmail.com and bouchannafa.k@gmail.com

Department of Mathematics and informatics, Polydisciplinary Faculty, Box 592, Sultan Moulay Slimane University, Beni Mellal, Morocco
e-mail : myabdallahidrissi@gmail.com

Received: September 5, 2022; Revised: March 15, 2023; Accepted: March 21, 2023

### Abstract

In this paper we consider a more general class of centralizers called I-centralizers. More precisely, given a prime ideal P of an arbitrary ring R we establish a connection between certain algebraic identities involving a pair of P-left centralizers and the structure of the factor ring R/P.

Keywords: Prime ring, Prime ideal, P-centralizer, Commutativity

### 1. Introduction

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 xy. Recall that an ideal P of R is prime if for all x,yR, xRyP implies xP or yP. An additive mapping d:RR is called a derivation if d(xy)=d(x)y+xd(y) holds for all x,yR. An additive mapping F:RR is called a generalized derivation if there exists a derivation d:RR such that F(xy)=F(x)y+xd(y) for all x,yR, 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:RR 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,yR. 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:RRRR 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 Qr (see [5, Chapter 2]). If R has the identity element then T:RR 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:RR an additive mapping.

• (1) f is called an I-left centralizer if f(xy)f(x)yI for all x,yR.

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

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

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

### 2. Identities Involving a Pair of Left P-Centralizers

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 PI. Suppose that T1 and T2 are two P-left centralizers on R, satisfying the condition T1(x)aaT2(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

[T1(x)aaT2(x),r]P  for all  r,xI.

Replacing x by xy in (2.1), we obtain

[T1(x)ya,r][aT2(x)y,r]P  for all  r,x,yI

in such a way that

(T1(x)aaT2(x))[y,r]+[T1(x)[y,a],r]P  for all  r,x,yI.

Substituting yr for y in (2.2), we get

[T1(x)y[r,a],r]P  for all  r,x,yI.

That is

T1(x)y[[r,a],r]+[T1(x),r]y[r,a]+T1(x)[y,r][r,a]P  for all  r,x,yI.

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

[T1(x),r]T1(x)y[r,a]P  for all  r,x,yI.

According to Fact 1.1, we obtain for each r∈ I, either[T1(x),r]T1(x)P or [r,a]P. Define A={rI/[T1(x),r]T1(x)Pfor allxI} and B={rI/[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 RII, then [R,a]P.

Now consider A=I, in this situation

[T1(x),r]T1(x)P  for all  r,xI.

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

[T1(x),s]rT1(x)P  for all  r,s,xI.

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

[T1(x),s]I[T1(x),s]P  for all  s,xI.

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

[r,aT2(x)]P  for all  r,xI

which means that

a[r,T2(x)]+[r,a]T2(x)P  for all  r,xI.

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

a[r,T2(x)]t+aT2(x)[r,t]+[r,a]T2(x)tP  for all  x,tI.

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

aT2(x)[r,t]P  for all  r,t,xI.

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

aT2(x)I[u,t]P  for all  t,u,xI.

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 aZ(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 PI. 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,yI;

• (2) T1(R)PandT2(R)P or R/P is a commutative integral domain.

Proof. By given assumption, we have

T1(xy)T2(yx)¯Z(R/P)  for all  x,yI.

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

[T2(y)[x,r],r]P  for all  r,x,yI.

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

T2(y)[T2(y)[x,r],r]+[T2(y),r]T2(y)[x,r]P  for all  r,x,yI.

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

[T2(y),r]T2(y)[x,r]P  for all  r,x,yI.

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

[T2(y),r]T2(y)t[x,r]P  for all  r,t,x,yI.

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

T1(x)[y,x]+[T1(x),x]yP  for all  x,yI.

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

T1(x)y[u,x]P  for all  u,x,yI.

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,yI;

• (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,yI;

• (2) T(xy)Z(R) for all x,yI;

• (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)xyZ(R) or T(xy)yxZ(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)±xyZ(R) for all x,yI;

• (2) T(xy)±yxZ(R) for all x,yI;

• (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 PI. 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

T1(x)T2(x)¯Z(R/P)  for all  xI.

A Linearization of (2.14) gives

T1(x)T2(y)+T1(y)T2(x)¯Z(R/P)  for all  x,yI.

This means that

[T1(x),r]T2(y)+T1(x)[T2(y),r]+T1(y)[T2(x),r]+[T1(y),r]T2(x)P                      for all  r,x,yI.

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

(T1(x)T2(y)+T1(y)T2(x))[T2(x),r]P  for all  r,x,yI.

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

(T1(x)T2(y)+T1(y)T2(x))t[T2(x),r]P  for all  r,t,x,yI.

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

T2(x)[s,r]P  for all  r,s,xI.

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)wwT2(x))P for all w,x,y ∈ I. Thereby obtaining,

T1(y)z(T2(x)wwT2(x))P  for all  w,x,y,zI.

Therefore, either T1(R)P or T2(x)wwT2(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 PI. 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,yI;

• (2) T1(x)T2(y)¯Z(R/P) for all x,yI;

• (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,yI, we suppose that

[T1(x),T2(y)]¯Z(R/P).

This may be rewritten as

[[T1(x),T2(y)],r]P  for all  r,x,yI.

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

[T1(x),T2(y)][t,r]+T2(y)[[T1(x),t],r]+[T2(y),r][T1(x),t]P.

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

[T1(x),T2(y)][T1(x),r]P  for all  r,x,yI.

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

[T1(x),T2(y)]I[T1(x),r]P  for all  r,x,yI.

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

I1={xI/[T1(x),T2(y)]Pfor allyI}andI2={xI/[T1(x),I]P}.

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,yI, then replacing y by ys, one obtains

T2(y)[T1(x),s]P  for all  s,x,yI.

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

T2(y)u[T1(x),s]P  for all  s,u,x,yI.

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

T1(x)[z,r]P  for all  x,y,zI.

Writing xw for x in (2.23), we get

T1(x)w[z,r]P  for all  r,w,x,zI.

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,yI;

• (2) T1(x)T2(y)Z(R) for all x,yI;

• (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 PI. 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,yI;

• (2) T1(x)T2(y)xy¯Z(R/P) for all x,yI;

• (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)xyZ(R) for all x,yI, 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,yI;

• (2) T1(x)T2(y)±xyZ(R) for all x,yI;

• (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 PPP=(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 PPP for all r,x,y ∈ I, therefore, Theorem 2.11 yields that for all PP, R/P is commutative which, because of PPP=(0), assures that R is commutative.

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