330 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 
 

 
Some Results on Double Centralizer for Prime and Semiprime Г- rings 

  

Aya Hussein Khudair 1* , Abdulrahman H. Majeed 2 ,  

Shrooq Bahjat Smeein 3   and Azza I.M.S. Abu-Shams 4   
1 Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. 

2 Department of Mathematic, Al-Mamoun University College, Baghdad, Iraq. 
3 Department of Information Technology -Section Mathematics, University of Technology and Applied 

Science - Muscat, Sultanate of Oman. 

 doi.org/10.30526/38.2.3594 

 
Abstract   

        The goal of this work, is to examine the concept of a double centralizer ( T, S ), and double 

Jordan  centralizer on prime and semiprime Г-rings, this is done by studying examples ,remarks 

and results related to that concepts  and looking for the  conditions under  which T equal S, we 

prove the results, the first result , let A be a semiprime Γ-ring and T is a left centralizer, S is a 

right centralizer, and they fulfilling  x 𝛼 T(y) = S (x) 𝛼 y,  for each  x ∈ A, 𝛼 ∈ Γ, thence (T,S) 

is a double centralizer. The second, let A be a prime Γ-ring, U be a not equal zero ideal of A, 

such that, T is a left centralizer, S is a right centralizer, and fulfilling x 𝛼T(y) = S (x) 𝛼 y, for 

each x, y ∈ U, 𝛼 ∈ Γ, thence (T, S) is a double centralizer. The third, let A be a prime Γ-ring, U 

be a not equal zero ideal of A and we get, if T=S on U, thence T=S on A. 

Keywords: Prime Г-rings, Semiprime Г-rings, centralizer, Jordan centralizer, double centralizer, 

double Jordan centralizer. 

 

1. Introduction 

        Barnes  (1) defined 𝛤-ring. Let A and Γ be two additive abelian groups. It there is a 

mapping (𝑥, 𝛼, 𝑦) → (𝑥 𝛼 𝑦) of 𝐴 ×  Г × 𝐴 → 𝐴, satisfying the following, for any 𝑥, 𝑦, 𝑧 ∈ A and 

α, β ∈𝛤.      

i.  (𝑥 + 𝑦)𝛼𝑧 = 𝑥𝛼𝑧 + 𝑦𝛼𝑧,   

         𝑥(α + β)y = xαy + xβy, 

         𝑥𝛼(y + z) = xαy + xαz, 

ii.  (𝑥𝛼𝑦)𝛽𝑧 = 𝑥𝛼(𝑦𝛽𝑧), thence A is named a 𝛤-ring. 

*Corresponding Author. 

 

 Received: 11 June 2023                        Accepted: 10 September 2023               Published: 20 April 2025  

4 Department of Mathematics, College of Science, Philadelphia University, Ammaan ,Jordan. 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0002-8259-4959
mailto:aya.hussein.g@gmail.com
https://orcid.org/0000-0001-8534-0749
mailto:dulrahman.h.majeed@almamonuc.edu.iq
https://orcid.org/0009-0002-9351-4176
mailto:shrooq.smeein@hct.edu.om
https://orcid.org/0009-0003-1741-1447
mailto:aabushams@philadelphia.edu.jo


IHJPAS. 2025, 38(2) 
 

331 
 

OZDEN et al. (2) defined the subring. A subring of Г-ring A is additive subgroup S of A such 

that  𝑆Г𝑆 ⸦ 𝑆 . Let A be a 𝛤-ring, thence A is named a commutative gamma-ring if, 𝑥α𝑦 = 𝑦𝛼𝑥,  

holds for any 𝑥, 𝑦 ∈  𝐴  and α ∈𝛤, Kandamar et al. (3). A subset 𝑈 of the 𝛤-ring A is a right (left) 

ideal of A if U is an additive subgroup of A and 𝑈Г𝐴 = {𝑎αx: a ∈ 𝑈, 𝛼 ∈  Γ, 𝑥 ∈ A} ( AГU)  is 

contained in U. If U is both a left and a right ideal, thence U is a two-sided ideal, or simply is an 

ideal of A. Barnes (1). A 𝛤-ring A is named prime if 𝑚Г𝐴Г𝑛 = (0) with 𝑚, 𝑛 ∈  A  implies 𝑚 =

0 𝑜𝑟 𝑛 = 0 and semiprime if  𝑚Г𝐴Г𝑚 = 0  with  𝑚 ∈ 𝐴 implies 𝑚 = 0 (4, 5). An ideal P of a 

gamma-ring A is prime ideal if for any ideals N,M⊆A, N𝛤M⊆ 𝑃  implies N⊆ 𝑃 or M⊆ 𝑃, Kyuno 

(5). A gamma-ring A is said to be prime 𝛤-ring if the zero ideal is prime ideal, Kyuno (5). Let A 

be a 𝛤-ring, thence A is named n-torsion free if  𝑛 𝑥 = 0, yields  𝑥 = 0, for every  𝑥 ∈ A, where  

𝑛 is positive integer, Chakraborty et al. (6). Let A be a gamma-semiring, an element 1 ∈ A , is 

named unity for any  𝑥 ∈  𝐴 there are  𝛼  ∈  𝛤 such that 𝑥 𝛼 1 =  1 𝛼 𝑥 =  𝑥, RAO (7).  Özkum 

et al. (8) defined the derivation and (Jordan derivation), let A be a gamma-ring and 𝐷 ∶ 𝐴 → A 

and additive map. Thence D is derivation (resp. Jordan derivation), if 𝐷 (𝑚 𝛼𝑛 )  =

 𝐷 (𝑚) 𝛼 𝑛 +  𝑚𝛼 𝐷 (𝑛) (resp. 𝐷 (𝑚 𝛼 𝑚) =  𝐷 (𝑚)𝛼 𝑚 +  𝑚𝛼 𝐷(𝑚)), for any  𝑚, 𝑛 ∈ A and 

α ∈ Г, Özkum et al. (8). Every derivation of A, is Jordan derivation but the   converse in 

general is not true, see Saleh (9). Barnes (1) defined the 𝛤-homomorphism. Let A and Y both be 

𝛤-rings, and ∅ a map of A in to A. Thence ∅ is a 𝛤-homomorphism, if and only 𝑖𝑓 ∅(𝑥𝛼𝑦) =

∅(𝑥)𝛼 ∅ (𝑦) , 𝑓𝑜𝑟 𝑎𝑙𝑙𝑥, 𝑦 ∈  𝐴 and  𝛼 ∈ Г. If ∅ is also one-to one and onto thence ∅ is a 𝛤-

isomorphism. An additive mapping ∅ of 𝛤-ring A into a 𝛤-ring A' is named Jordan 

homomorphism if ∅(𝑥𝛼𝑦 + 𝑦𝛼𝑥) = ∅(x)α∅(𝑦) + ∅(y)α∅(𝑥), for each  𝑥, 𝑦 ∈ A  and 𝛼 ∈ Г, 

Shaheen (10). Let A be a 𝛤-ring, a mapping d of A, to itself is named 𝛤-centralizing on a subset 

𝑆 of A if [ 𝑥, 𝑑 (𝑥)]𝛼  ∈ Z(A), for every 𝑥 ∈  𝑆 and α  ∈ Г,in the special case when  [ 𝑥, 𝑑 (𝑥)]𝛼 =

0, hold for any 𝑥 ∈ S and 𝛼 ∈ Г, the mapping d is named 𝛤-commuting on 𝑆 , Sameer et al. (11). 

Many researchers have studied centralizers and derivations in prime and semiprime 𝛤- rings (12-

21) and (22-30). The objective of this paper is to debate, double centralizer ( T, S ), and double 

Jordan  centralizer  on prime and semiprime 𝛤- rings, with fulfilling certain identities. 

 

2. Preliminaries and Fundamentals   

2.1 Definition 

Ali et al. (18) Let A be a gamma-ring, for any 𝑥, 𝑦 ∈ A  and  𝛼 ∈ Г, the symbol [𝑟, 𝑡]𝑎 =

𝑟 α 𝑡 ‒  𝑡 𝛼 𝑟, to symbolize the commutator. 𝑇(𝑟   ⃘ 𝑡) = 𝑟 α 𝑡 +  𝑡 𝛼 𝑟. 

2.2 Lemma  

Ali et al. (18) If A is a gamma-ring, for any 𝑟, 𝑡, 𝑠 ∈ A and α, 𝛽 ∈ Г thence:  

I. [𝑟, 𝑡]α  +  [𝑡, 𝑟]α  = 0    

II. [𝑟 + 𝑡, 𝑠] α  = [𝑟, 𝑠]α+[𝑡, 𝑠]α 

III. [𝑟, 𝑡 + 𝑠]α= [𝑟, 𝑡]α+[𝑟, 𝑠]α 

IV.  [𝑟, 𝑡]α+β= [𝑟, 𝑡]α+[𝑟, 𝑡]ᵦ 

V. [𝑟𝛽𝑡 , 𝑠]α  =  𝑟𝛽[𝑡, 𝑠]α +  [𝑟, 𝑠]α 𝛽𝑡 + 𝑟 𝛽 𝑠 α t ‒ 𝑟 α s 𝛽 𝑡 . 

2.3 Definition  

Hoque 20(19) An additive mapping 𝑇: A → A is a left (right)centralizer, if 𝑇(𝑟𝛼 𝑡 )  = 

𝑇 (𝑟) 𝛼 𝑡 ( 𝑇 (𝑟 𝛼 𝑡 )  =  𝑟 𝛼 𝑇 (𝑡)) holds for any 𝑟 , 𝑡 ∈ A and  𝛼 ∈ Г. A centralizer is both a left 

and right centralizer. 



IHJPAS. 2025, 38(2) 
 

332 
 

2.4 Example  

Let F be a field, and 𝐷2 (𝐹)be a diagonal matrices 2 by 2 over F and Γ = { [
0 0
0 𝑛

] , 𝑛 ∈ 𝑍}, 

define 𝑇: 𝐷2 (𝐹) → 𝐷2 (𝐹)  as   

𝑇 ([
𝑎 0
0 𝑏

])  = [
0 0
0 𝑏

]  , for any 𝑎, 𝑏 ∈ 𝐹. 

Thence T is a centralizer.  

2.5 Definition  

Hoque (19)An additive mapping 𝑇: 𝐴 →  𝐴, is Jordan left (right) centralizer, if 𝑇(𝑥α x) =

T(𝑥)𝛼𝑥 (𝑇(𝑥𝛼𝑥) = 𝑥𝛼𝑇(𝑥),for any 𝑥 ∈ A and  α ∈ Г. 

2.6 Definition  

Let A be gamma-ring, let  𝑇, 𝑆: 𝐴 → 𝐴, be an additive mappings, thence a mate ( T, S ) is named 

a double centralizer, if 𝑇 is a left centralizer, S is a right centralizer, and they satisfy a balanced 

requirement,  𝑥α T(𝑦) = 𝑆(𝑥) 𝛼𝑦  , for any 𝑥, 𝑦 ∈ 𝐴,   𝛼 ∈ Г. 

2.7 Definition  

Let A be gamma-ring, and let 𝑇, 𝑆: 𝐴 →  𝐴, be an additive mapping, thence a mate ( T, S ) is 

named a double Jordan centralizer, if T is a left Jordan centralizer, S is a right Jordan centralizer, 

and they satisfy a balanced requirement,  (𝑥𝛼 𝑇(𝑥) = 𝑆(𝑥) 𝛼 𝑥), for any 𝑥 ∈ 𝐴, 𝛼 ∈ Г. 

 

3. Main Results 

     In the following, we give the definition of commuting double centralizer: 

3.1 Definition  

Let A be a 𝛤-ring, and (T, S), be a double centralizer. Thence (T, S), is named commuting double 

centralizer, if T and S are commuting. 

Now, we shall give an example for a commuting double centralizer. 

3.2 Example  

  Let F be a field, and A be a 𝛤-ring of all triangular matrices of the from 

 𝑥 = {[

𝑑
0
0
0

   

𝑎
𝑑
0
0

   

𝑐
0
𝑑
0

  

𝑏
𝑐

−𝑎
𝑑

 ] , 𝑓𝑜𝑟𝑎𝑙𝑙 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐹},   and  Г = { [

0
0
0
0

   

0
0
0
0

   

0
0
0
0

  

𝑛
0
0
0

 ] , 𝑓𝑜𝑟𝑎𝑙𝑙 𝑛 ∈ Z} . 

In connection to the frequent process of addition and multiplication and let T, S: A→A be additive 

mappings defined by 

𝑇(𝑥) = 𝑦𝛼𝑥  and 𝑆(𝑥) = 𝑥𝛼𝑦, for each   𝑥, 𝑦 ∈ A and 𝛼 ∈ Г. 

Where; 

𝑦 = [

0
0
0
0

   

0
0
0
0

   

0
0
0
0

  

𝑏
0
0
0

 ] , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑏 ∈  F.  

It is clear that T and S are commuting double centralizer. 

In the following results, we give some certain conditions to obtain (T,S) is a double centralizer, 

where T and S are from A to A. 

3.3 Theorem 

  Let A be a semiprime 𝛤-ring and T, S: A→ A be a mapping fulfilling. 

𝑥𝛼𝑇(𝑦) = 𝑆(𝑥)𝛼𝑦, for each  𝑥, 𝑦 ∈ A and α ∈ Г.                                                                        (1) 

Thence (T, S) is a double centralizer. 



IHJPAS. 2025, 38(2) 
 

333 
 

Proof:  We need to show that T, S are additive mapping, and  

𝑇 (𝑥 𝛼 𝑦) =  𝑇 (𝑥) 𝛼 𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤.   

𝑆 (𝑥 𝛼 𝑦) =  𝑥 𝛼 𝑆 (𝑦), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝐴, 𝛼 ∈ 𝛤. 

Now  replace   𝑦   by   𝑦 + z  in (1), we imply  

𝑥𝛼 𝑇(𝑦 + 𝑧) = 𝑆(𝑥)𝛼𝑦 + 𝑆 (𝑥)𝛼𝑧,  for each 𝑥, 𝑦, 𝑧 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤.  

Hence  

𝑥𝛼( 𝑇(𝑦 + 𝑧) −  𝑇 (𝑦) − 𝑇(𝑧)) = 0, for 𝑎𝑙𝑙 𝑥, 𝑦, 𝑧 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤.   

By the semiprimeness of A, we imply  

𝑇(𝑦 + 𝑧)  =  𝑇(𝑦) +  𝑇 (𝑧), for each   𝑦, 𝑧 ∈  𝐴. 

Similarly, we can show that  

𝑆(𝑥 + 𝑦) = 𝑆 (𝑥) + 𝑆 (𝑦), for each  𝑥, 𝑦, ∈  𝐴. 

Now, replacing 𝑦 with 𝑦𝛽𝑧 in (1) we obtain 

𝑥 𝛼 (𝑇 (𝑦 𝛽𝑧) − 𝑇 (𝑦) 𝛽𝑧)  = 0, for each  𝑥, 𝑦, 𝑧 ∈ 𝐴 𝑎𝑛𝑑 𝛼, 𝛽 ∈ 𝛤.   

By the semiprimeness of A, we imply  

𝑇 (𝑦 𝛽 𝑧)  =  𝑇 (𝑦) 𝛽 𝑧, for each   𝑦, 𝑧 ∈  𝐴 𝑎𝑛𝑑 𝛽 ∈ 𝛤. 

Similarly, we can show  

𝑆 (𝑥 𝛼 𝑦)  =  𝑥 𝛼 𝑆 (𝑦) , for each 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤. 

Thence (T, S) is a double centralizer. 

Now, we give some results which make T=S under different conditions, where (T,S) is double 

centralizer. 

3.4 Theorem  

Let A be a prime 𝛤-ring, U be a not equal zero ideal of A. Let 𝑇, 𝑆: 𝐴 → 𝐴 be additive mappings 

such that T is a left centralizer, S is a right centralizer and they gratify 𝑥𝛼𝑇(𝑦) = 𝑆(𝑥)𝛼𝑦, for 

each 𝑥, 𝑦 ∈ 𝑈 𝑎𝑛𝑑 𝛼 ∈ Г. Thence (T,S) is a double centralizer. 

Proof:   We have 

𝑥 𝛼 𝑇(𝑦)  =  𝑆 (𝑥) 𝛼 𝑦, for each  𝑥, 𝑦 ∈  𝑈, 𝛼 ∈  𝛤.                                                                   (2) 

Replace x with 𝑥𝛽𝑟  in (2) when  𝑥 ∈ U, β ∈ Г 𝑎𝑛𝑑 𝑟 ∈ 𝐴, we imply   

𝑥 𝛽 (𝑟 𝛼 𝑇(𝑦) − 𝑆 (𝑟) 𝛼 𝑦) = 0, for each 𝑟 ∈ 𝐴, 𝑥, 𝑦 ∈  𝑈𝑎𝑛𝑑 𝛼 , 𝛽 ∈ 𝛤.    

i.e. 

𝑥 𝛾 𝐴 𝛽 (𝑟 𝛼 𝑇(𝑦) −  𝑆 (𝑟)𝛼 𝑦) = 0 , for each  𝑟 ∈ 𝐴, 𝑥, 𝑦 ∈ 𝑈𝑎𝑛𝑑 𝛼 , 𝛽, 𝛾 ∈  𝛤. 

By primeness of A and since U be a not equal zero ideal of A, we imply  

𝑟𝛼𝑇(𝑦) = 𝑆(𝑟)𝛼𝑦, for each 𝑟 ∈ 𝐴,   𝑦 ∈ 𝑈 𝑎𝑛𝑑 𝛼 ∈ 𝛤.                                                             (3) 

Replacing 𝑦 with 𝑡𝜎𝑦 in (3), where 𝑡 ∈  𝐴, 𝑦 ∈  𝑈, and σ ∈ Г. 

 (𝑟 𝛼 𝑇(𝑡) −  𝑆 (𝑟)𝛼 𝑡 )𝜎 𝑦 = 0 , for each  t, 𝑟 ∈ 𝐴,   𝑦 ∈ 𝑈 𝑎𝑛𝑑 𝛼, 𝜎 ∈  𝛤. 

Implies that  

(𝑟𝛼𝑇(𝑡) − S(r)αt)𝜎U𝛿𝐴 = 0, for each 𝑡, 𝑟 ∈ A and 𝛼, 𝜎, δ ∈ Г. 

By the primeness of A, we imply 

𝑟 𝛼 𝑇(𝑡) =  𝑆 (𝑟)𝛼 𝑡, for each 𝑡, 𝑟 ∈  𝐴, 𝑎𝑛𝑑, 𝛼 ∈  𝛤. 

3.5 Theorem  

Let A be a prime gamma-ring, U be a not equal zero ideal of A, and (T,S) be a double centralizer. 

If T=S on U, thence T=S on A. 

Proof: We have      

𝑇 (𝑥) =  𝑆 (𝑥), for each   𝑥 ∈  𝑈.                                                                                                 (4) 

By replacing  𝑥 with  𝑟𝛼𝑥  in (4), when  𝑟 ∈ A ,  𝑥 ∈ U  and 𝛼 ∈ Г , we imply  



IHJPAS. 2025, 38(2) 
 

334 
 

𝑇(𝑟)𝛼 𝑥 = 𝑟 𝛼 𝑆(𝑥) =  𝑟𝛼𝑇(𝑥 ), for each ∈ 𝐴, 𝑥 ∈ 𝑈 𝑎𝑛𝑑 𝛼 ∈ 𝛤 .                                           (5) 

Since (T, S) are a double centralizer, (5) leads to  

𝑇(𝑟)𝛼 𝑥 = 𝑆 (𝑟)𝛼 𝑥, for each 𝑥 ∈ 𝑈, 𝑟 ∈ 𝐴, 𝑎𝑛𝑑 𝛼 ∈  𝛤.i.e.      

(𝑇 (𝑟) − 𝑆 (𝑟))𝛼𝑈𝛽𝐴 = 0, for each 𝑟 ∈ A, and 𝛼, 𝛽 ∈ Г. 

Since A is a prime 𝛤-ring and U be a not equal zero ideal of A, we imply T = S. 

From Theorem above, we imply the following: 

3.6 Corollary  

Let A be a prime gamma-ring, U be an ideal of A and (T, S) be a double centralizer. If,  

T = S = 0 on U, thence T = S = 0 on A. 

In the following theorem, we shall prove that T=S in case T acts as a homomorphism on A. 

3.7 Theorem   

Let A be a semiprime gamma-ring and let (T,S) be a double centralizer, if T acts as a 

homomorphism on A, thence T = S. 

Proof: We have  

𝑇(𝑥𝛼𝑦) = 𝑇(𝑥)𝛼 𝑦 , for each 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑  𝛼 ∈ Г. 

Since T is acts homomorphism on A, thence  

T (x) 𝛼 T(y) = T (x) 𝛼 y, for any  𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤.                                                                 (6) 

On the other hand; 

𝑥 𝛼 𝑇(𝑦)  =  𝑆 (𝑥) 𝛼 𝑦 , for 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤.                                                                  (7) 

The substation 𝑇(𝑥) for 𝑥 in (7), gives  

𝑇(𝑥) 𝛼 𝑇(𝑦) = 𝑆(𝑇(𝑥)) 𝛼 𝑦, for each 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤.                                                        (8) 

By comparing (6) with (8), we arrive at (𝑆(𝑇(𝑥)) − 𝑇(𝑥)) 𝛼 𝑦 = 0 

Multiply from the right by (𝑆(𝑇(𝑥)) − 𝑇(𝑥)), we get 

(𝑆(𝑇(𝑥)) − 𝑇(𝑥))𝛽 𝑦 𝛼 (𝑆(𝑇(𝑥)) − (𝑇(𝑥)) = 0, for each  𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼, 𝛽 ∈ Г. 

By semiprimeness of A, we have  

𝑆 (𝑇(𝑥)) = 𝑇(𝑥), for each 𝑥 ∈ 𝐴.                                                                                                (9) 

From (9) and using (7), we imply 

𝑇(𝑥)𝛼𝑇(𝑦) = 𝑇(𝑥)𝛼𝑆(𝑦), for each  𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈  𝛤.                                                       (10) 

Replace 𝑥  by 𝑥𝛽𝑧  and  𝑦  by  𝑦𝜎𝑤 in (7) and using (10), we arrive at  

𝑥𝛽𝑧 𝛼𝑇(𝑦)𝜎𝑆 (𝑤) = 𝑆(𝑥 𝛽 𝑧) 𝛼 𝑦 𝜎 𝑤, for each  𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝐴 𝑎𝑛𝑑 𝛼, 𝛽, 𝜎, ∈ 𝛤.                  (11) 

Thence by (11), we imply  

𝑥𝛽𝑆(𝑧)𝛼 𝑦 𝜎 𝑆 (𝑤) = 𝑆 (𝑥 𝛽 𝑧)𝛼𝑦𝜎𝑤, for 𝑎𝑙𝑙 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝐴 𝑎𝑛𝑑  𝛼, 𝛽, 𝜎 ∈ 𝛤. 

Whence it follows that  

𝑥𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤) − w) = 0, for each  𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝐴𝑎𝑛𝑑 𝛼 , 𝛽, 𝜎 ∈ 𝛤.                                   (12) 

The substitution on S (w)− w for x in (12), gives us  

( 𝑆(𝑤) − 𝑤)𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤) − 𝑤) = 0, for each  𝑦, 𝑧, 𝑤 ∈ 𝐴 𝑎𝑛𝑑 𝛼, 𝛽, 𝜎 ∈ Γ. 

Right multiplication the above relation by S(z), yields  

(𝑆(𝑤) − 𝑤)𝛽𝑆(𝑧)𝛼𝑦𝜎(𝑆(𝑤) − 𝑤)𝛾𝑆(𝑧) = 0, for 𝑎𝑙𝑙 𝑦, 𝑧, 𝑤 ∈ 𝐴 𝑎𝑛𝑑 𝛼, 𝛽, 𝜎, 𝛾 ∈ Γ. 

By the semiprimeness of A, we obtain 

(𝑆(𝑤) − w)βS(𝑧) = 0, for each  𝑤, 𝑧 ∈ A and β ∈ Г. 

This gives  

𝑆(𝑤) 𝛽 𝑆(𝑧) =  𝑤 𝛽 𝑆(𝑧), for each 𝑤, 𝑧 ∈  𝐴 𝑎𝑛𝑑 𝛽 ∈ 𝛤.                                                        (13) 

From (7) and (13), we obtain  



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𝑥 𝛼 𝑇(𝑦)  =  𝑆(𝑥) 𝛼 𝑆(𝑦)  =  𝑥 𝛼 𝑆(𝑦), for each 𝑥, 𝑦 ∈  𝐴 𝑎𝑛𝑑  𝛼 ∈  Г. 

Of course, we have also,  

𝑥𝛼(𝑇(𝑦) −S(y)) = 0, for each  x, y ∈ A, and  𝛼 ∈ Г. 

By the semiprimeness of A, we imply T=S. 

In the following theorem, we gives a relation between T and S, where (T,S) is a double centralizer 

on prime Γ-ring. 

3.8 Theorem   

Let A be a prime 𝛤-ring, and U be a not equal zero ideal of A, we imply (T,S) be a double 

centralizer. If 𝑇(𝑟 𝛼 𝑥) =  𝑆(𝑟)𝛼𝑥 𝑓𝑜𝑟 𝑎𝑙𝑙  𝑟 ∈ 𝐴, 𝑥 ∈  𝑈, 𝑡ℎ𝑒𝑛 𝑇 = 𝑆. 

Proof:  We have  

𝑇(𝑟𝛼𝑥) = 𝑇(𝑟)𝛼𝑥 = 𝑆(𝑟)𝛼𝑥   for each 𝑟, 𝑡 ∈ A , 𝑥 ∈ U and 𝛼 ∈ Г.    

This reduces to  

(𝑇(𝑟) − 𝑆(𝑟))𝛼𝑥 = 0, for each  𝑟 ∈ A ,  𝑥 ∈ U and 𝛼 ∈ Г.                                                       (14) 

Replacing x with 𝑡𝛽𝑥 in (14), when  𝑡 ∈ A, 𝑥 ∈ U, 𝛼 ∈ Г, leads to 

 (𝑇(𝑟) − 𝑆(𝑟))𝛼𝑡𝛽𝑥 = 0, for each  𝑟 ∈  𝐴, 𝑥 ∈  𝑈 𝑎𝑛𝑑 𝛼, 𝛽 ∈ Г. 

i.e. 

(𝑇(𝑟)  − 𝑆(𝑟))𝛼𝐴 𝛽𝑈 =, for each  𝑟 ∈ A and 𝛼 , 𝛽 ∈ Г. 

Since A is a prime gamma-ring, and U be a not equal zero ideal, we have T = S. 

3.9 Theorem   

Let A be a prime 𝛤-ring, and (T, S) be a double centralizer, if T acts as a not equal zero Jordan 

homomorphism (𝑇(𝑥𝛼𝑥) = 𝑇(𝑥)𝛼 𝑇(𝑥))  for each 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤. Thence T=S=id. 

Proof : We have  

𝑇(𝑥𝛼𝑥) = 𝑇(𝑥)𝛼𝑥, for each  𝑥 ∈ A and  𝛼 ∈ Г.  

Thence from above relation and since T is acts as Jordan homomorphism. 

Yields, 

𝑇(𝑥) 𝛼 (𝑇(𝑥) –  𝑥)  = 0, for each 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ Г.                                                                 (15) 

Replace 𝑥  by 𝑥𝛽 𝑦 in (15), we imply 

 𝑇(𝑥)𝛽 𝑦 𝛼 (𝑇(𝑥)–  𝑥)𝛽 𝑦 = 0, for each  𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼, 𝛽 ∈ 𝛤.                                              (16) 

Linearization (16), we imply 

 (𝑇(𝑥) 𝛽 𝑦 𝛼 ( 𝑇(𝑥) –  𝑥)𝛽 𝑧 +  𝑇(𝑥) 𝛽 𝑧 𝛼( 𝑇(𝑥) –  𝑥)𝛽 𝑦 = 0                                             (17) 

Now, replacing 𝑧 by  𝑦𝜎𝑧 in (17) and using (16), we obtain 

(𝑇(𝑥) 𝛽 𝑦 𝛼 𝐴 𝛾 ( 𝑇(𝑥) –  𝑥)𝛽𝑦, for each   𝑥, 𝑦 ∈  𝐴, 𝛼, 𝛽, 𝛾 ∈ Г.   

By the primeness of A, we imply 

𝑇(𝑥)  =  𝑥, for each 𝑥 ∈ A.                                                                                                         (18) 

Otherwise, T=0. From 𝑥𝛼𝑇(𝑦) = S(x)𝛼𝑦, and by (18), we imply T = S = id. 

 

4. Conclusion 

      This work is to discuss double centralizer ( T, S ), and double Jordan  centralizer  on prime 

and semiprime Г- rings, with fulfilling certain identities. We prove that; when T is a left 

centralizer, S is a right centralizer, and they fulfilling 𝑥𝛼𝑇(𝑦) = 𝑆(𝑥)𝛼𝑦, for each 𝑥, 𝑦 ∈

𝑈 𝑎𝑛𝑑 𝛼 ∈ Г. Then (T,S) is a double centralizer. Also if (T, S) be a double centralizer,  T acts as 

a homomorphism on A, then T=S, and  if T acts as a not equal zero Jordan 

homomorphism (𝑇(𝑥𝛼𝑥) = 𝑇(𝑥)𝛼 𝑇(𝑥))  for each 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤. Then T = S = id.  



IHJPAS. 2025, 38(2) 
 

336 
 

 

Acknowledgment 

        Our researcher extends his Sincere thanks to the editor and members of the preparatory 

committee of the Ibn AL-Haitham Journal of Pure and Applied Sciences. 

 

Conflict of Interest  

        There are no conflicts of interest. 

 

Funding  

        There is no funding for the article. 

 

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