EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5493
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Jordan φ-Centralizers on Semiprime and Involution
Rings

Abu Zaid Ansari1, Faiza Shujat2,∗, Alwaleed Kamel1,3, Ahlam Fallatah2

1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, K.S.A.
2 Department of Mathematics, Faculty of Science, Taibah University, Madinah, K.S.A.
3 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82749, Egypt

Abstract. The intention of the current investigation is to demonstrate that if an additive mapping
H : R → R fulfills certain identities, thenH is a φ-centralizer on R, where R is any suitable, torsion-
free semiprime ring and p is a fixed integer greater than or equal to 1. As a result of the primary
theorems, involution Iv related observations are also provided. We will also consider criticism and
discussion alongside the proofs of theorems. Suitable examples are given in favor of justification.

2020 Mathematics Subject Classifications: AMS 16N60, 16W10, 16R50, 47B47

Key Words and Phrases: Algebraic identities, Semiprime ring, (Jordan) φ-centralizer

1. Introduction

In order to effectively comprehend our concept, we must first recall a few fundamental
ideas. Throughout, R shall stand for an associative ring with unity e. A ring R is termed
as p-torsion free, where p > 1, if pr = 0 entails r = 0 for every r ∈ R. A ring R is
recognised as a prime if rRt = {0} implies that either r = 0 or t = 0, and is termed as a
semiprime if rRr = {0} yields r = 0. The study of Helgosen [5], who introduced the idea
of centralizers on Banach algebras, which is also known as multipliers, is supportive of our
current understanding.

A possible idea of centralizers on commutative Banach algebra put out by Wang [14].
Further study on centralizers for topological algebras and the continuity of centralizers
on Banach algebras is done by Johnson [9]. Further, Johnson studiedthe behaviour of
centralizers on algebra of compact operators on Banach space over itself in [7]. Since,
every centralizer on a commutative faithful Banach algebra is continuous. Simultaneously,
he studied another class of Banach algebras that includes the group of algebras of locally

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5493

Email addresses: ansari.abuzaid@gmail.com (A. Z. Ansari), faiza.shujat@gmail.com (F. Shujat),
wld kamel22@yahoo.com (A. Kamel), afallatah@taibahu.edu.sa (A. Fallatah)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 2 of 10

compact topological groups in [8].

Additionally, Husain [6] has studied centralizers on topological algebras, specifically
focusing on topological algebras with orthogonal bases and full metrizable locally con-
vex algebras. Authors have since examined centralizers and double centralizers on certain
topological algebras in [10, 12]. In [2], authors investigated the idea of Jordan ∗-derivations
on standard operator algebras. The study of Hopf algebras, representation theory of Ba-
nach algebras, the study of Banach modules, the theory of singular integrals, interpola-
tion theory, stochastic processes, the theory of semigroups of operators, partial differential
equations, and the study of approximation problems are just a few of the fields in which
centralizers have also been used (for more information, see Larsen [11]).

An additive mapping H : R → R is said to be a left (right) centralizer if it holds
H(rt) = H(r)t

(
respectively, H(rt) = rH(t)

)
for all r, t ∈ R and it is known as a Jordan

right (Jordan left) centralizer if H(r2) = rH(r)
(
respectively, H(r2) = H(r)r

)
for all

r ∈ R. In recognizing that this mapping H is both a right centralizer and a left central-
izer, we refer to it as a centralizer. As a result of Albas [1], H : R → R is called as a left
(right) φ-centralizer if H(rt) = H(r)φ(t)

(
H(rt) = φ(r)H(t)

)
and it is additive, for all

r, t ∈ R and is known as a Jordan right (Jordan left) φ-centralizer if H(r2) = φ(r)H(r)(
H(r2) = H(r)φ(r)

)
for all r ∈ R, where φ is an endomorphism on R. An additive map-

ping H is recognised as a φ-centralizer, if H is both left as well as right φ-centralizer.
Every Jordan φ-centralizer is a φ-centralizer. However, this isn’t usually the case. Under
appropriate torsion restrictions, the converse of this statement is also true for a semiprime
ring provided in [1].

Motivated by previous literature review, in the present paper, authors presented an
extension of this mathematical statement. Specifically, H : R → R is a φ-centralizer, if H
fulfills any one of the following 3H(r3p) = H(rp)φ(r2p) + φ(rp)H(rp)φ(rp) + φ(r2p)H(rp),
2H(r2p) = H(rp)φ(rp) + φ(rp)H(rp) and H(r3p) = φ(rp)H(rp)φ(rp) for every r in a
semiprime ring R that is specifically torsion restricted.

2. On φ-centralizer

The following outcome is required to validate the basic theorems:

Lemma 1 ([4, Theorem 1.2]). If φ is a surjective endomorphism on a semiprime ring
R with 2 torsion free condition and H : R → R is an additive mapping that satisfies
2H(r2) = H(r)φ(r) + φ(r)H(r) for all r ∈ R. Then H is a φ-centralizer on R.

We start with the study considering the following problem:

Theorem 1. If φ is a surjective endomorphism on a semiprime ring R with (3p − 1)!
torsion free condition and H : R → R is an additive mapping that satisfies

3H(r3p) = H(rp)φ(r2p) + φ(rp)H(rp)φ(rp) + φ(r2p)H(rp) for all r ∈ R, (1)



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 3 of 10

then H is a φ-centralizer on R, where p is a fixed integer greater than or equal to 1.

Proof. We commence with the equation (1) by replacing r by r + kt, we get the fol-
lowing for k being a positive integer and t ∈ R

3H(r3p+
(
3p
1

)
(r3p−1)kt+

(
3p
2

)
r3p−2k2t2+ ...+k3pt3p) = H(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+

...+kptp)·φ(r2p+
(
2p
1

)
r2p−1kt+

(
2p
2

)
r2p−2k2t2+...+k2pt2p)+φ(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+

...+kptp) ·H(rp+
(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+ ...+kptp) ·φ(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+ ...+

kptp)+φ(r2p+
(
2p
1

)
r2p−1kt+

(
2p
2

)
r2p−2k2t2+ ...+k2pt2p) ·H(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+

...+ kptp).

Restate the preceding expression using (1) as

kA1(r, t) + k2A2(r, t) + ...+ k3p−1A3p−1(r, t) = 0,

where the coefficients of ki are delimited by Ai(r, t) for all i = 1, 2, ..., (3p− 1). A system
of (3p− 1) homogeneous equations can be obtained if we substitute 1, 2, ..., (3p− 1) for k
one by one, it provides a Vandermonde matrix

1 1 · · · 1
2 22 · · · 23p−1

· · · · · ·
· · · · · ·
· · · · · ·

3p− 1 (3p− 1)2 · · · (3p− 1)3p−1

 .

Which yields that Ai(r, t) = 0 for all r, t ∈ R and for i = 1, 2, .., (3p−1). In particular,
We have A1(r, t) = 0 implies that

3
(
3p
1

)
H(r3p−1t) =

(
2p
1

)
H(rp)φ(r2p−1t) +

(
p
1

)
H(rp−1t)φ(r2p)

+
(
p
1

)
φ(rp−1t)H(rp)φ(rp) +

(
p
1

)
φ(rp)H(rp−1t)φ(rp)

+
(
p
1

)
φ(rp)H(rp)φ(rp−1t) +

(
2p
1

)
φ(r2p−1t)H(rp)

for each r, t ∈ R. If we put e in place of r in above expression, then we find 9pH(t) =
2pH(e)t + 2ptH(e) + 3pH(t) + pH(e)t + ptH(e), for each t ∈ R. Making use of torsion
restrictions on R to obtain

2H(t) = H(e)φ(t) + φ(t)H(e), for all t ∈ R. (2)

Next, A2(r, t) = 0 implies that

3
(
3p
2

)
H(r3p−2t2) =

(
2p
2

)
H(rp)φ(r2p−2t2) +

(
p
1

)(
2p
1

)
H(rp−1t)φ(r2p−1t)

+
(
p
2

)
H(rp−2t2)φ(r2p) +

(
p
2

)
φ(rp)H(rp)φ(rp−2t2)

+
(
p
1

)(
p
1

)
φ(rp−1t)H(rp)φ(rp−1t) +

(
p
1

)(
p
1

)
φ(rp)H(rp−1t)φ(rp−1t)

+
(
p
2

)
φ(rp)H(rp−2t2)φ(rp) +

(
p
1

)(
p
1

)
φ(rp−1t)H(rp−1t)φ(rp)

+
(
p
2

)
φ(rp−2t2)H(rp)φ(rp) +

(
p
2

)
φ(r2p)H(rp−2t2)

+
(
2p
1

)(
p
1

)
φ(r2p−1t)H(rp−1t) +

(
2p
2

)
φ(r2p−2t2)H(rp).



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 4 of 10

Reword the above expression by putting e in place of r, we have

33p(3p−1)
2 H(t2) = 2p(2p−1)

2 H(e)φ(t2) + 2p2H(t)φ(t) + p(p−1)
2 H(t2)

+p(p−1)
2 H(e)φ(t2) + p2φ(t)H(e)φ(t) + p(p−1)

2 H(t2)

+p2φ(t)H(e)φ(t) + p2φ(t)H(t) + p(p−1)
2 φ(t2)H(e)

+p(p−1)
2 H(t2) + 2p2φ(t)H(t) + 2p(2p−1)

2 φ(t2)H(e)

Simplify the above expression to find

9p(3p− 1)H(t2) = 2p(2p− 1)H(e)φ(t2) + 4p2H(t)φ(t) + p(p− 1)H(t2)
+p(p− 1)H(e)φ(t2) + 2p2H(t)φ(t) + p(p− 1)H(t2)
+2p2φ(t)H(e)φ(t) + 2p2φ(t)H(t) + p(p− 1)φ(t2)H(e)
+p(p− 1)H(t2) + 4p2φ(t)H(t) + 2p(2p− 1)φ(t2)H(e)

Collect the like terms in above expression and make them more comprehensible as

(24p2 − 6p)H(t2) = (5p2 − 3p)2H(t2) + 6p2(H(t)φ(t)
+φ(t)H(t)) + 2p2φ(t)H(e)φ(t),

for all t ∈ R. Again comparing the like terms both side and using equation (2), we come
up with

14p2H(t2) = 6p2(H(t)φ(t) + φ(t)H(t)) + 2p2φ(t)H(e)φ(t),

for all t ∈ R. Hence, we get by applying the torsion freeness of R

14H(t2) = 6
(
H(t)φ(t) + φ(t)H(t)

)
+ 2φ(t)H(e)φ(t), for every t ∈ R. (3)

Multiplying from right side by φ(r) to (2), we obtain

2H(r)φ(r) = H(e)φ(r2) + φ(r)H(e)φ(r) for all r ∈ R.

Multiply from left side by φ(r) to (2) to get

2φ(r)H(r) = φ(r)H(e)φ(r) + φ(r2)H(e) for all r ∈ R.

Adding these equations, we find

2(H(r)φ(r) + φ(r)H(r)) = 2φ(r)H(e)φ(r) + 2H(r2),

which implies that

2φ(r)H(e)φ(r) = 2(H(r)φ(r) + φ(r)H(r))− 2H(r2) for all r ∈ R.

Using this equation in (3), we have

14H(r2) = 6
(
H(r)φ(r) + φ(r)H(r)

)
+ 2

(
H(r)φ(r) + φ(r)H(r)

)
− 2H(r2) for all r ∈ R.



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 5 of 10

Using torsion restrictions on R, we get

2H(r2) = H(r)φ(r) + φ(r)H(r) for all r ∈ R.

Using Lemma 1, we obtained the required outcome.

Theorem 2. If φ is a surjective endomorphism on a semiprime ring R with (2p − 1)!
torsion free condition and H : R → R is an additive mapping that satisfies

2H(r2p) = H(rp)φ(rp) + φ(rp)H(rp) for all r ∈ R, (4)

then H is a φ-centralizer on R, where p is a fixed integer greater than or equal to 1.

Proof. We proceed with (4) and replace r by r + kt for k ≥ 1 and t ∈ R, to get

2H(r2p+
(
2p
1

)
(r2p−1)kt+

(
2p
2

)
r2p−2k2t2+ ...+k2pt2p) = H(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+

...+kptp) ·φ(rp +
(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2 + ...+kptp) +φ(rp +

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2 +

...+ kptp) · H(rp +
(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2 + ...+ kptp) for all r, t ∈ R and k ≥ 1.

Rewrite the above expression by using (4) as

kQ1(r, t) + k2Q2(r, t) + ...+ k2p−1Q2p−1(r, t) = 0.

A system of (2p − 1) homogeneous equations can be identified if k is substituted by
1, 2, ..., (2p − 1) in turn. It provides a Vandermonde matrix of type (2p − 1) × (2p − 1),
then for i = 1, 2, .., (2p− 1), Qi(r, t) = 0 for all r, t ∈ R. Particularly, Q1(r, t) = 0 suggests

2
(
2p
1

)
H(r2p−1kt) =

(
p
1

)
H(rp)φ(rp−1t) +

(
p
1

)
H(rp−1)φ(trp)

+
(
p
1

)
φ(rp)H(rp−1t) +

(
p
1

)
φ(rp−1t)H(rp)

(5)

Reinstate the above equation by putting e in place of r to have 4pH(t) = pH(e)φ(t) +
pH(t) + pH(t) + pφ(t)H(e). On simplifying the last relation we can obtain 2pH(t) =
pH(e)φ(t) + pφ(t)H(e) for all t ∈ R. A torsion restriction given in the hypothesis enable
us to write

2H(t) = H(e)φ(t) + φ(t)H(e), for all t ∈ R. (6)

Now consider the following Q2(r, t) = 0, we have

2
(
2p
2

)
H(r2p−2t2) =

(
p
2

)
H(rp)φ(rp−2t2) +

(
p
1

)(
p
1

)
H(rp−1t)φ(rp−1t)

+
(
p
2

)
H(rp−2t2)φ(rp) +

(
p
2

)
φ(rp)H(rp−2t2)

+
(
p
1

)(
p
1

)
φ(rp−1t)H(rp−1t) +

(
p
2

)
φ(rp−2t2)H(rp)

Substitute e for r in above expression to obtain

22p(2p−1)
2 H(t2) = p(p−1)

2 H(e)φ(t2) + p2H(t)φ(t) + p(p−1)
2 H(t2)

+p(p−1)
2 H(t2) + p2φ(t)H(t) + p(p−1)

2 φ(t2)H(e)



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 6 of 10

A simple manipulation yields that 2p2H(t2) = p2(H(t)φ(t) + φ(t)H(t)). By utilizing the
torsion-freeness of R, we achieved 2H(t2) = H(t)φ(t) +φ(t)H(t) for all t ∈ R. Hence H is
carry oneself like φ-centralizer, as desired.

Theorem 3. If φ is a surjective endomorphism on a semiprime ring R with (3p − 1)!
torsion free condition and H : R → R is an additive mapping that satisfies

H(r3p) = φ(rp)Hφ((rp))rp for all r ∈ R. (7)

Then H is a φ-centralizer on R, where p is a fixed integer greater than or equal to 1.

Proof. Replacing r by r + kt in (7), we obtain

H(r3p+
(
3p
1

)
(r3p−1)kt+

(
3p
2

)
r3p−2k2t2+...+k3pt3p) = φ(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+...+

kptp) ·H(rp+
(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+ ...+kptp)φ(rp+

(
p
1

)
rp−1kt+

(
p
2

)
rp−2k2t2+ ...+kptp),

where k is a positive integer.

Rewrite the above expression by using (7) as

kR1(r, t) + k2R2(r, t) + ...+ k3p−1R3p−1(r, t) = 0,

where Ri(r, t) signifies for the coefficients of power of k up to (3p − 1). Replacing k
by 1, 2, ..., (3p − 1) in turn, we obtain a system of (3p − 1) homogeneous equations that
provides a Vander Monde matrix of (3p − 1) by (3p − 1) that implies that for all r, t ∈
R, Ri(r, t) = 0, where i = 1, 2, .., (3p − 1). In particular, We have

(
3p
1

)
H(r3p−1t) =(

p
1

)
φ(rp)H(rp−1t)φ(rp) +

(
p
1

)
φ(rp)H(rp)φ(rp−1t)−

(
p
1

)
φ(rp−1t)H(rp)φ(rp) for all r, t ∈ R.

Replacing r by e and making use of torsion restriction on R to get

2H(t) = H(e)φ(t) + φ(t)H(e), for all t ∈ R. (8)

Further, R2(r, t) = 0 implies that

(
3p
2

)
H(r3p−2t2) =

(
p
2

)
φ(rp)H(rp)φ(rp−2t2) +

(
p
1

)(
p
1

)
φ(rp−1t)H(rp)φ(rp−1t)

+
(
p
1

)(
p
1

)
φ(rp)H(rp−1t)φ(rp−1t) +

(
p
2

)
φ(rp)H(rp−2t2)φ(rp)

+
(
p
1

)(
p
1

)
φ(rp−1t)H(rp−1t)φ(rp) +

(
p
2

)
φ(rp−2t2)H(rp)φ(rp).

Reword the above expression by putting e in place of r, we have

3p(3p−1)
2 H(t2) =

(
p
2

)
H(e)φ(t2) +

(
p
1

)(
p
1

)
φ(t)H(e)φ(t)

+
(
p
1

)(
p
1

)
H(t)φ(t) +

(
p
2

)
H(t2)

+
(
p
1

)(
p
1

)
φ(t)H(t) +

(
p
2

)
φ(t2)H(e).

Simplify the above expression using the same steps as we did in last theorems to find
3p2H(t2) = p2(H(t)φ(t) + φ(t)H(t)) + p2φ(t)H(e)φ(t), for all t ∈ R. Hence, we get by
applying the torsion freeness of R



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 7 of 10

3H(t2) = (H(t)φ(t) + φ(t)H(t)) + φ(t)H(e)φ(t), for all t ∈ R. (9)

Multiplying from right side by φ(r) to (8), we obtain

2H(r)φ(r) = H(e)φ(r2) + φ(r)H(e)φ(r) for every r ∈ R.

Multiply from left side by φ(r) to (8) to get

2φ(r)H(r) = φ(r)H(e)φ(r) + φ(r2)H(e) for every r ∈ R.

Adding these equations, we find

2(H(r)φ(r) + φ(r)H(r)) = 2φ(r)H(e)φ(r) + 2H(r2),

which implies that

φ(r)H(e)φ(r) = H(r)φ(r) + φ(r)H(r)−H(r2) for every r ∈ R.

Using this equation in (9), we have

4H(r2) = 2(H(r)φ(r) + φ(r)H(r)) for every r ∈ R.

Using torsion restrictions on R, get 2H(r2) = H(r)φ(r)+φ(r)H(r) for every r ∈ R. Using
Lemma 1, H is a φ-centralizer on R.

The subsequent illustration supports our theorems:

Example 1. Let R =
{(

z1 0
0 z2

)
| z1, z2 ∈ 2Z8

}
, whereby the meaning of Z8 is as

usual. Define mappings H, φ : R → R by H(r) =

(
0 0
0 z2

)
and φ(r) =

(
z2 0
0 z1

)
for all r ∈ R. Clearly, R is not a semiprime ring but H satisfy the algebraic identity of
main theorems ((ii) p > 1) of this section. It is easy to see that H is not a φ-centralizer.
Therefore semiprimeness hypothesis is crucial for the above theorems.

3. On involution ring

Next, an additive mapping Iv : R → R is said to be an involution if it satisfies
Iv(rt) = Iv(t)Iv(r) and Iv(Iv(r)) = r for all r, t ∈ R. A ring with involution is a ring that
possesses an involution Iv. An additive mappingH : R → R is termed as a right (resp. left)
Iv-centralizer if H(rt) = Iv(r)H(t) (resp. H(rt) = H(r)Iv(t)) holds for all r, t ∈ R and H is
termed as a right (resp. left) Jordan Iv-centralizer if for all r ∈ R, H(r2) = Iv(r)H(r) (resp.
H(r2) = H(r)Iv(r)). If H is both right and left (Jordan) Iv-centralizer then, recognised
as a (Jordan) Iv-centralizer of R. Let φ be an endomorphism on R. An additive mapping



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 8 of 10

H : R → R is known as a right (resp. left) φ-Iv-centralizer if H(rt) = φ(Iv(r))H(t) (resp.
H(rt) = H(r)φ(Iv(t))) holds for all r, t ∈ R and it is φ-Iv-centralizer if it left as well right
φ-Iv-centralizer. An additive mapping H is a right (resp. left) Jordan φ-Iv-centralizer if
for all r ∈ R, H(r2) = φ(Iv(r))H(r) (resp. H(r2) = H(r)φ(Iv(r))). If it is both, then it is
Jordan φ-Iv-centralizer of R.

We investigate about the interesting extension of the results presented in the last sec-
tion in context of involution ring. The study of some identities on involution ring is the
main focus of this section. However, evidence suggests that an additive mappings H on
a suitable torsion free restricted semiprime ring R satisfying 2H(r2p) = H(rp)φ(Iv(r))

p +
φ(Iv(r))

pH(rp), 3H(r3p) = H(rp)φ(Iv(r))
2p + φ(Iv(r))

pH(rp)φ(Iv(r))
p + φ(Iv(r))

2pH(rp)
and H(r3p) = φ(Iv(r))

pH(rp)φ(Iv(r))
p for all r ∈ R, will be a φ-Iv-centralizer of R.

We require the following lemma that supports our primary findings.

Lemma 2 ([3, Corollary 2.1]). If φ is a surjective endomorphism on a 2 torsion free
semiprime ring R with involution Iv and H : R → R is an additive mapping that satisfies
2H(r2) = H(r)φ(Iv(r)) + φ(Iv(r))H(r) for all r ∈ R, then H is a φ-Iv-centralizer on R.

Next, start main result of this part.

Theorem 4. If φ is a surjective endomorphism on a (3p−1)! torsion free semiprime ring
R with involution Iv and H : R → R is an additive mapping that satisfies any one of the
following algebraic identities:

(i) 3H(r3p) = H(rp)φ(Iv(r))
2p + φ(Iv(r))

pH(rp)φ(Iv(r))
p + φ(Iv(r))

2pH(rp)

(ii) 2H(r2p) = H(rp)φ(Iv(r))
p + φ(Iv(r))

pH(rp)

(iii) H(r3p) = φ(Iv(r))
pH(rp)φ(Iv(r))

p for all r ∈ R,

then, H is a φ-Iv-centralizer on R, where p is a fixed integer greater than or equal to 1.

Proof. (i) Define a mapping T : R → R such that T (r) = H(Iv(r)) for all r ∈ R.
Then,

T (r + s) = H(Iv(r + s))
= H(Iv(r) + Iv(s))
= H(Iv(r)) +H(Iv(s))
= T (r) + T (s)

for every r, s ∈ R. Therefore, T is an additive mapping on R. Now, consider

3T (r3p) = 3H(Iv((r
3p))

= H((Iv(r))
p)φ((r)2p) + φ((r)p)H((Iv(r))

p)φ((r)p) + φ((r)2p)H(Iv(r))
= T (rp)φ(r2p) + φ(rp)T (rp)φ(rp) + φ(r2p)T (rp) for all r, t ∈ R.



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 9 of 10

Use first theorem of the last section to find that T is a φ-centralizer on R. Hence,

H(r2) = H(Iv(Iv(r
2))

= T (Iv(r
2)

= T (Iv(r)Iv(r))
= T (Iv(r))φ(Iv(r)), as T is a φ-centralizer on R
= H(r)φ(Iv(r))

This implies that H is a Jordan left φ-Iv-centralizer on R. Similarly, we can prove that
H is a Jordan right φ-Iv-centralizer on R. Now, combine and use Lemma 2 to obtain the
required conclusion.

Using same technique, we can get proof of (ii) and (iii) part.

Example 2. Let R =
{(

z1 0
0 z2

)
| z1, z2 ∈ 2Z8

}
is a ring with involution Iv : R →

R by Iv(r) =

(
z2 0
0 z1

)
, where Z8 signifies what it normally does. Define mappings

H, φ : R → R by H(r) =

(
0 0
0 z2

)
and ϕ(r) =

(
2z2 0
0 2z1

)
for all r ∈ R. It is

clear that H satisfy the identities in Theorem 4 ((ii) p > 1) and R is neither a 2-torsion
free semiprime ring nor H is a φ-Iv-centralizer on R, hence semiprimeness hypothesis is
crucial for Theorem 4.

4. Conclusion

We explore φ-centralizer on rings and φ-Iv-centralizer on rings with involution in depth,
which is an intriguing topic. Obtaining continuity theorems on other algebraic structures,
such as Banach algebra, semi-simple Banach algebra, Lie algebra, C∗ algebra, etc., is the
area of future study in the framework of the provided research. It would be fascinating
to view our concept in the context of [13] with the aid of algebra of linear operators
(transformations). It is also interesting that the reader can consider various functional
identities involving specific sorts of derivations, such as generalized (α, β)-derivations on
semiprime rings with involution and generalized (α, β)-higher derivations. The forms of
additive maps applying to rings and their corresponding subsets have been described using
just algebraic techniques.

Competing Interests

Regarding the publication of this work, the authors affirm that they have no conflicts
of interest.



A. Z. Ansari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5493 10 of 10

Acknowledgements

The authors are extremely grateful to the reviewers and editor for their generous
suggestions, insightful remarks and recommendations to make this manuscript well orga-
nized. Authors extend their gratitude to the Deanship of Higher Education and Scientific
Research at the Islamic University of Madinah for the support provided to the Post-
Publishing Program.

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