EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6044
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On generalized (α, ∗)-derivations and α-centralizers on
Rings

Faiza Shujat1,∗, Salwa Alharbi1

1 Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia

Abstract. The intention of the current research is to define the concept of (α, ∗)-derivations on
ring R, where α is an automorphism of R and ∗ represents involution on R. We obtain some
commutativity theorems in case of prime ring by utilizing the role of α and ∗. We will also
discuss the proofs of theorems in case of non-commutative prime ring and under which condition
generalized (α, ∗)-derivation behaves like an α-centralizers. Suitable examples are given in favor of
introduced concept.

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

Key Words and Phrases: Generalized (α, ∗)-derivations, Prime ∗-ring, α-centralizer

1. Introduction

Through out the manuscript, the notation Z(R) stands for the center of an associative
ring R. The symbol [b, d] specifies the commutator of b, d ∈ R, which is represented by
the mathematical formula bd − db. If pr = 0 implies r = 0 for every r ∈ R and p > 1 is
a fixed integer, then a ring R is a p-torsion free ring. A ring R is a prime if rRt = {0}
gives that either t = 0 or r = 0. It is called semiprime if it fulfills the requirement that
cRc = {0} yields that c = 0.

In simple terms, the mapping ζ is (skew)-commuting on R if ζ(c)c + cζ(c) = 0 for
each of c ∈ R. If ζ(c)c + cζ(c) ∈ Z(R) for each c ∈ R, then a map ζ from R to R is
thought to be (skew)-centralizing on R. If the mapping η from R to R fulfills the equation
η(ce) = η(c)e+ cη(e), for each of c, e ∈ R, then it is regarded as a derivation on R.

Let R be a ring whose automorphism is β. A map h on R satisfying h(dk) =
h(d)β(k) + dh(k) is recognized as the β-derivation (skew-derivation) if it holds for any

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6044

Email addresses: faiza.shujat@gmail.com, fullahkhan@taibahu.edu.sa (F. Shujat),
salwa.alharbi1990@gmail.com (S. Alharbi)

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



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 2 of 11

pair d, k in R and h has additivity. The combination form h = β − I served as the β-
derivation if we symbolize the identity map on R by I.

Before moving onto this section’s primary findings, we clarify a few fundamental con-
cepts and terminologies. Involution is an additive mapping defined as ∗ from R to R that
fulfills the following two requirements: (dj)∗ = j∗d∗ and (d∗)∗ = d for each d, j ∈ R. In-
vertible matrices and identity matrices are the most common examples of involution over
the matrix ring. A ∗-ring, sometimes referred to as an involution ring (or ring combined
with an involution ∗). The references [1], [2], [3], [4], [5], [6] are ideal places for start
reading about generalized derivations, involution, centralizers and their related topics.

A ring possessing involution ∗ is called a ∗-prime ring if aRb = aRb∗ = {0}, or
aRb = a∗Rb = {0}, where a, b ∈ R, implies that either a = 0 or b = 0. It is a noticeable
fact that all prime rings possessing involution ∗ are ∗-prime but not necessarily prime. For
example, R0 denotes the opposite ring of a prime ring R, then R × R0 having exchange
involution ∗xe defined as

∗xe(x, y) = (y, x)

is a ∗xe-prime but not prime.

Let R be a ∗-ring. A mapping D : R → R is said to be a ∗-derivation if it satisfies:
(i) Additivity and (ii) D(xy) = D(x)y∗ + xD(y) for all x, y ∈ R. In the case where R
is a commutative ∗-ring, D has the form D(x) = a(x − x∗) for some a ∈ R, which is a
∗-derivation on R.

Following [7], a mapping T : R → R is called a left (right) centralizer if T (xy) = T (x)y
(T (xy) = xT (y)) holds for all x, y ∈ R and T is also additive. In the same line of in-
vestigation, the expression for ∗-centralizer comes out as follows: A mapping T on R,
additive and satisfying T (xy) = T (x)y∗ and T (xy) = x∗T (y) for all x, y ∈ R will be called
left ∗-centralizer and right ∗-centralizer respectively on R. A remarkable investigation on
the theory of centralizers and ∗-centralizers presented in [8–11]. In [4], authors proved an
advancement of the generalized concept of ∗-derivation on standard operator algebra.

A ring with endomorphism α, if we take γ = ς − α, then γ is an (α, I)-derivation, but
not a derivation when R is semiprime and I = α. The inclusive information can be found
in [12]. Some commutativity results about ∗-bimultipliers and generalized ∗-biderivations
can be viewed in [3]. We review the concept of such γ and introduce the concept of (α, ∗)-
derivation and generalized (α, ∗)-derivation on R as follows:

Definition 1. Let D : R −→ R be an additive map. D is said to be (α, ∗)-derivation on
R if it satisfy the conditions:

D(νk) = D(ν)α(k) + ν∗D(k), for every ν, k ∈ R,



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 3 of 11

where ∗ is an involution on R and α is the automorphism on R.

Example 1. Consider a ∗-ring R =
{(

p 0
q r

)
| p, q, r ∈ 2Z4

}
. Define involution

mapping ∗ from R to itself by

(
p 0
q r

)∗
=

(
−p 0
0 0

)
for all p ∈ 2Z4 under matrix

addition and matrix multiplication, where Z4 has its usual notation. Take a mapping

α,D : R → R defined by α

[(
p 0
q r

)]
=

(
r 0
q p

)
and D

[(
p 0
q r

)]
=

(
0 0
q 0

)
for

all p, q, r ∈ 2Z4. It is clear that D satisfy the above definition, hence it is (α, ∗)-derivation
on R.

We observe that if ∗ = I, the definition 1 will be set as skew derivation with automor-
phism α. It is somewhat a unified notion of skew derivation and ∗-derivation. Next we
extend our definition to the case of generalized derivation.

Definition 2. Let F,D : R −→ R be additive maps. F is said to be generalized (α, ∗)-
derivation associated with D on R if it satisfy the below condition

F(νk) = F(ν)α(k) + ν∗D(k), for every ν, k ∈ R,

where ∗ is an involution on R and α is the automorphism on R.

Example 2. Let

R =


0 a b
0 0 c
0 0 0

 ∣∣∣∣∣ a, b, c ∈ 2Z4


be a ring with usual operation of matrix addition and multiplication.

Define F,D : R → R as

F(r) =

0 0 b
0 0 0
0 0 0

 ,

and a map D is given by

D(r) =

0 0 0
0 0 c
0 0 0

 , r ∈ R.

Define α : R → R by

α(r) =

0 −a −b
0 0 −c
0 0 0

 .

The involution is given by

r∗ =

0 c b
0 0 a
0 0 0

 .

F will be a generalized (α, ∗)-derivation associated with D.



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 4 of 11

A lot of research has been done in the context of involution involved with derivation,
generalized derivation, Jordan derivation, left derivation, etc. Our present research is mo-
tivated by all the above theories and the role of automorphism on R and involution. We
will prove some commutativity theorems in the setting of prime and semiprime rings. We
will observe that α and ∗ play a crucial role in our proofs. The commutativity theorem on
prime rings possessing automorphisms (or endomorphisms) proved in [5, 6, 9, 13]. Further
we refer the reader to the extensive bibliography contained in it.

Motivated by the above literature review and concepts, we putout the extension of the
notion of generalized (α, ∗)-derivation to generalized (α, ∗)-n-derivation on rings as follows:

Definition 3. A mapping F : Rn → R is called a generalized (α, ∗)-n -derivation if there
exists an (α, ∗)-n-derivation D : Rn → R such that

F (ς1, . . . , ςkς
′
k, . . . , ςn) = F (ς1, . . . , ςk, . . . , ςn)α(ς

′
k) + ς∗kD(ς1, . . . , ς

′
k, . . . , ςn)

for all ς1, . . . , ςk, ς
′
k, . . . , ςn ∈ R.

2. Results on prime ∗-ring

We start with the following results:

Lemma 1. [14] The center of R includes the center of a nonzero ideal (one-sided) for a
semiprime ring R. Any commutative ideal (one-sided) is immediately enclosed Z(R).

Theorem 1. Let a semiprime ∗-ring be R, ∗ be an involution, and α be an automorphism
on R. If F1,F2 are two generalized (α, ∗)-n-derivations on R associated with (α, ∗)-n-
derivation D1,D2 respectively, such that

F1(ς1, ς2, ..., ςn) = F2(ς1, ς2, ..., ςn)

for all ς1, ς2, ..., ςn ∈ R, then D1 = D2.

Proof. By the hypothesis, we are given that

F1(ς1, ς2, ..., ςn) = F2(ς1, ς2, ..., ςn), for every ς1, ς2, ..., ςn ∈ R. (1)

Rewrite (1) to get the form

F1(ς1, ..., ςkς
′
k, ..., ςn) = F2(ς1, ..., ςkς

′
k, ...,n ), for every ς1, ς2, ..., ςk, ς

′
k, ..., ςn ∈ R. (2)

By definition the above equation reword as

F1(ς1, . . . , ςk, . . . , ςn)α(ς
′
k) + ς∗kD1(ς1, . . . , ς

′
k, . . . , ςn) = F2(ς1, . . . , ςk, . . . , ςn)α(ς

′
k)

+ς∗kD2(ς1, . . . , ς
′
k, . . . , ςn).



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 5 of 11

Application of (1) with the last expression to find

ς∗kD1(ς1, . . . , ς
′
k, . . . , ςn) = ς∗kD2(ς1, . . . , ς

′
k, . . . , ςn)

= ς∗k (D1(ς1, . . . , ς
′
k, . . . , ςn)−D2(ς1, . . . , ς

′
k, . . . , ςn))

= 0, for each ς1, . . . , ςk, . . . , ςn ∈ R.
(3)

Particularly consider ςk = ς∗k for the case of ∗ − ring to obtain

ςkR
(
D1(ς1, . . . , ς

′
k, . . . , ςn)−D2(ς1, . . . , ς

′
k, . . . , ςn)

)
= 0 for each ς1, . . . , ςk, . . . , ςn ∈ R.

Condition of semiprimeness of R implies that

D1(ς1, . . . , ς
′
k, . . . , ςn) = D2(ς1, . . . , ς

′
k, . . . , ςn), for every ς1, . . . , ς

′
k, . . . , ςn ∈ R

Therefore, D1 = D2. This completes the proof.

Theorem 2. If a prime ∗-ring R admits a nonzero (α, ∗)-n-derivation D, then R is
commutative.

Proof. Given that D is a (α, ∗)-n-derivation, by definition we have

D(ς1, . . . , ςkς
′
k, . . . , ςn) = D(ς1, . . . , ςk, . . . , ςn)α(ς

′
k) + ς∗kD(ς1, . . . , ς

′
k, . . . , ςn)

for every ςk, ς
′
k ∈ R. Now substitute ςk = ςky, where y ∈ R, we get

D(ς1, . . . , ςkyς
′
k, . . . , ςn) = D(ς1, . . . , ςky, . . . , ςn)α(ς

′
k) + (ςky)

∗D(ς1, . . . , ς
′
k, . . . , ςn)

for every ςk, ς
′
k, y ∈ R. Expanding the terms,

D(ς1, . . . , ςkyς
′
k, . . . , ςn) = D(ς1, . . . , ςk, . . . , ςn)α(y)α(ς

′
k)

+ ς∗kD(ς1, . . . , y, . . . , ςn)α(ς
′
k)

+ y∗ς∗kD(ς1, . . . , ς
′
k, . . . , ςn)

(4)

for every ς1, .., ςk, ς
′
k, ..., ςn, y ∈ R. Alternative expression for the left hand side of above

equation is given by

D(ς1, . . . , ςk(yς
′
k), . . . , ςn) = D(ς1, . . . , ςk, . . . , ςn)α(yς

′
k)

+ ς∗kD(ς1, . . . , yς
′
k, . . . , ςn),

for every ς1, . . . , ςk, ς
′
k, . . . , ςn, y, ς

∗
k ∈ R. Expanding the terms

D(ς1, . . . , ςk(yς
′
k), . . . , ςn) = D(ς1, . . . , ςk, . . . , ςn)α(y)α(ς

′
k)

+ ς∗kD(ς1, . . . , y, . . . , ςn)α(ς
′
k)

+ ς∗ky
∗D(ς1, . . . , ς

′
k, . . . , ςn).

(5)

Substituting equation (4) into equation (5), we get

0 = (ς∗ky
∗ − y∗ς∗k)D(ς1, . . . , ς

′
k, . . . , ςn). for every ς1, ..., ςn, y ∈ R.



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 6 of 11

This simplifies to
0 = [ς∗k , y

∗]D(ς1, . . . , ς
′
k, . . . , ςn).

Now, substitute ς∗k = ςk and y∗ = y into the above equation,

[ςk, y]D(ς1, . . . , ς
′
k, . . . , ςn) = 0 for everyς1, . . . , ς

′
k, . . . , ςn ∈ R.

Replace y by yr to obtain

[ςk, y]rD(ς1, . . . , ς
′
k, . . . , ςn) = 0 for every ς1, . . . , ς

′
k, . . . , ςn, y, r ∈ R. (6)

By primeness, we conclude that either

D(ς1, . . . , ς
′
k, . . . , ςn) = 0 for every ς1, . . . , ς

′
k, . . . , ςn ∈ R

or
[ςk, y] = 0 for every ςk, y ∈ R.

Since D(ς1, . . . , ς
′
k, . . . , ςn) ̸= 0, it follows that

[ςk, y] = 0 for every ςk, y ∈ R.

Thus, R is commutative.

Corollary 1. If a non-commutative prime ∗-ring R admits a (α, ∗)-n-derivation D, then
D =0.

Theorem 3. Let R be a prime ∗-ring. If R admits a nonzero generalized (α, ∗)-n-
derivation F associated with an (α, ∗)-n-derivation D, then one of the conditions hold:
1. R is commutative.
2. F acts as left α-centralizer.

Proof. Since F is a generalized (α, ∗)-n-derivation, then we obtain for all
ς1, ..., ςk, ς

′
k, ..., ςn ∈ R

F(ς1, ..., ςkyς
′
k, ..., ςn) = F(ς1, . . . , ςky, . . . , ςn)α(ς

′
k) + (ςky)

∗D(ς1, . . . , ς
′
k, . . . , ςn). (7)

Simplify above expression to get

F(ς1, ..., ςkyς
′
k, ..., ςn) = F(ς1, . . . , ςk, . . . , ςn)α(y)α(ς

′
k)

+(ςk)
∗D(ς1, . . . , y, . . . , ςn)α(ς

′
k)

+y∗ς∗kD(ς1, . . . , ς
′
k, . . . , ςn)

(8)

for every ς1, ..., ςk, y, ς
′
k, ..., ςn in R. Alternatively in view of (7) we find

F (ς1, . . . , ςkyς
′
k, . . . , ςn) = F (ς1, . . . , ςk, . . . , ςn)α(yς

′
k) + ς∗kD(ς1, . . . , yς

′
k, . . . , ςn). Expand

the right hand side of last expression to get

F (ς1, . . . , ςk, . . . , ςn)α(y)α(ς
′
k)+ ς∗kD(ς1, . . . , y, . . . , ςn)α(ς

′
k)+ ς∗ky

∗D(ς1, . . . , ς
′
k, . . . , ςn) (9)



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 7 of 11

Combining (15) and (9) together, we get for every ς1, ..., ςk, y, ς
′
k, ..., ςn in R

y∗ς∗kD(ς1, . . . , ς
′
k, . . . , ςn) = ς∗ky

∗D(ς1, . . . , ς
′
k, . . . , ςn) (10)

Put ςk and y instead of ς∗k and y∗ to observe that

[ςk, y](D(ς1, . . . , ς
′
k, . . . , ςn)) = 0 for every ς1, ..., ςk, y, ς

′
k, ..., ςn ∈ R. (11)

Again replace y by yr where r ∈ R in (11) and using it

[ςk, y]R(D(ς1, . . . , ς
′
k, . . . , ςn)) = 0 for every ς1, ..., ςk, y, ς

′
k, ..., ςn ∈ R. (12)

From (12), we say that R can be written as K+
1 ∪K+

2 , where

K+
1 = {[ςk, y] = 0 | ςk, y ∈ R}

and
K+

2 = {ς1..., ςn ∈ R | D(ς1, . . . , ς
′
k, . . . , ςn) = 0}.

Which is a contradiction to the fact that R cannot be determined by the union of two
additive subgroups, namely K+

1 and K+
2 . Hence, primeness implies that either K+

1 = R
or K+

2 = R.
If K+

1 = R, then R is commutative by Lemma 1. In case K+
2 = R, we say that after

simple manipulation
[D(ς1, . . . , ςn), r] = 0 for every r ∈ R.

Hence, D commutes with R. An application of Theorem 2 guarantees that either D = 0
or R is commutative. Again, we are done in the second case. On the other hand take
D = 0, and use the definition to get the expression

F (ς1, . . . , ςkr, . . . , ςn) = F (ς1, . . . , ςk, . . . , ςn)α(r)

for every ς1, . . . , ςk, . . . , ςn, r ∈ R, where F acts as a left α-centralizer.

Theorem 4. Let R be a non-commutative prime ∗-ring. If R admits a nonzero generalized
(α, ∗)-n-derivation F associated with an (α, ∗)-n-derivation D, then F acts as left α-
centralizer.

Proof. The proof is straight forward by the application of Theorem 3.

Theorem 5. Let R be a 2-torsion-free prime ∗-ring having generalized (α, ∗)-n-derivations
F1 and F2 associated with (α, ∗)-n-derivations D1 and D2 respectively. If

F1(ς1, . . . , ςk, . . . , ςn)D2(y1, . . . , yk, . . . , yn)
−F2(ς1, . . . , ςk, . . . , ςn)D1(y1, . . . , yk, . . . , yn) = 0,

for each ς1, ..., ςn, y1, ..., yn ∈ R, then one of the following holds:



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 8 of 11

(i) F1 = 0 or F2 acts as a left α-centralizer.

(ii) F2 = 0 or F1 acts as a left α-centralizer.

Proof. Suppose that

F1(ς1, . . . , ςk, . . . , ςn)D2(y1, . . . , yk, . . . , yn)
−F2(ς1, . . . , ςk, . . . , ςn)D1(y1, . . . , yk, . . . , yn) = 0.

(13)

Put ykz in place of yk, we have for each ς1, ..., ςn, y1, ..., yn ∈ R

F1(ς1, . . . , ςk, . . . , ςn)D2(y1, . . . , ykz, . . . , yn)
−F2(ς1, . . . , ςk, . . . , ςn)D1(y1, . . . , ykz, . . . , yn) = 0.

Explore the above equation

F1ς1, . . . , ςk, . . . , ςn){D2(y1, . . . , yk, . . . , yn)α(z) + y∗kD2(y1, . . . , z, . . . , yn)
}

− F2(ς1, . . . , ςk, . . . , ςn){D1(y1, . . . , yk, . . . , yn)α(z) + y∗kD1(y1, . . . , z, . . . , yn)
}
= 0.

From (13), we arrive at

F1(ς1, . . . , ςk, . . . , ςn)y
∗
kD2(y1, . . . , z, . . . , yn)

−F2(ς1, . . . , ςk, . . . , ςn)y
∗
kD1(y1, . . . , z, . . . , yn) = 0,

(14)

for each ς1, ..., ςn, y1, ..., yn ∈ R. Multiplying (14) from the right by
pD1(y

′
1, . . . , y

′
k, . . . , y

′
n) where p, y′k ∈ R, we obtain

(F1(ς1, . . . , ςk, . . . , ςn)y
∗
kD2(y1, . . . , z, . . . , yn)

−F2(ς1, . . . , ςk, . . . , ςn)y
∗
kD1(y1, . . . , z, . . . , yn))pD1(y

′
1, . . . , y

′
k, . . . , y

′
n) = 0,

(15)

for each ς1, ..., ςn, y1, ..., yn in R.
Case 1 In view of (14), (15) takes the form

2F2(ς1, . . . , ςk, . . . , ςn)ykD1(y1, . . . , z, . . . , yn)pD1(y
′
1, . . . , y

′
k, . . . , y

′
n) = 0

Using ∗-primeness of R and 2-torsion-freeness of R, we find

F2(ς1, . . . , ςk, . . . , ςn)ykD1(y1, . . . , z, . . . , yn) = 0, for every ς1, ..., ςn, y1, ..., yn, z ∈ R.

Again, making use of primeness, we can conclude either F2 = 0 or D1 = 0. In case D1 = 0,
we obtain

F1(ς1, . . . , ςkς
′
k, . . . , ςn) = F1(ς1, . . . , ςk, . . . , ςn)α(ς

′
k),

which implies that F1 acts as an α-centralizer on R.

Case 2 Multiply (15) by pD2(y
′
1, . . . , y

′
k, . . . , y

′
n) from the right and use it again to

obtain

2F1(ς1, . . . , ςk, . . . , ςn)ykD2(y1, . . . , z, . . . , yn)pD2(y
′
1, . . . , y

′
k, . . . , y

′
n) = 0,

for each ς1, ..., ςn, y1, ..., yn inR. In view of (15), the equation takes the form After repeating
the similar footsteps as in case 1, we conclude either F1 = 0 or D2 = 0. In case D2 = 0,
F2 acts as a left α-centralizer.



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 9 of 11

Theorem 6. Let R be a semi-prime ∗-ring admitting a generalized (α, ∗)-n-derivation F
with associated (α, ∗)-n-derivation D. Then

D(R,R, . . . , R) ⊆ Z(R).

Proof. Since R is a ∗-ring and admits a generalized (α, ∗)-n-derivation F , we take a
quick start from equation (6) in Theorem 2

[x, y]RD(ς1, . . . , ς
′
k, . . . , ςn) = 0 for every ς1, ..., ςn ∈ R. (16)

Replacing x by D(ς1, . . . , ς
′
k, . . . , ςn)x, we obtain for every ς1, ..., ςn ∈ R

D(ς1, . . . , ς
′
k, . . . , ςn)[x, y]RD(ς1, . . . , ς

′
k, . . . , ςn)

+[D(ς1, . . . , ς
′
k, . . . , ςn), y]xRD(ς1, . . . , ς

′
k, . . . , ςn) = 0.

(17)

From (16) and (17), we conclude

[D(ς1, . . . , ς
′
k, . . . , ςn), y]xRD(ς1, . . . , ς

′
k, . . . , ςn) = 0 for every ς1, ..., ςn ∈ R. (18)

we can write also

[D(ς1, . . . , ς
′
k, . . . , ςn), y]xRyD(ς1, . . . , ς

′
k, . . . , ςn) = 0 for every ς1, ..., ςn, y, x ∈ R. (19)

Now multiply (18)by y from right and subtract the resulting equation with (19)to find

[D(ς1, . . . , ς
′
k, . . . , ςn), y]xR[D(ς1, . . . , ς

′
k, . . . , ςn), y] = 0 for every ς1, ..., ςn, x, y ∈ R.

Since this holds for all y ∈ R, it follows that

[D(ς1, . . . , ς
′
k, . . . , ςn), y] = 0 for every y ∈ R.

Thus, we conclude that

D(ς1, . . . , ς
′
k, . . . , ςn) ⊆ Z(R).

Theorem 7. Let R be a semiprime ring with involution ∗. If F is a generalized (α, ∗)-n-
derivation of R associated with an (α, ∗)-n-derivation D such that

F (ς1, . . . , ςk, . . . , ςn)yi = ςiF (y1, . . . , yk, . . . , yn)

for every ς1, . . . , ςk, . . . , ςn, y1, . . . , yk, . . . , yn ∈ R, then F is a left α-centralizer.



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 10 of 11

Proof. By the given hypotheses, we have

F (ς1, . . . , ςk, . . . , ςn)yi = ςiF (y1, . . . , yk, . . . , yn) (20)

for every ς1, . . . , ςk, . . . , ςn, y1, . . . , yk, . . . , yn ∈ R. Substituting yk = ykz, where z ∈ R, we
obtain

F (ς1, . . . , ςk, . . . , ςn)yi = ςiF (y1, . . . , ykz, . . . , yn).

Using definition on the right hand side, we have

F (ς1, . . . , ςk, . . . , ςn)yi = ςiF (y1, . . . , yk, . . . , yn)α(z) + ςiy
∗
kD(y1, . . . , z, . . . , yn).

Since α is an automorphism, we may put z = α−1(z) to have

F (ς1, . . . , ςk, . . . , ςn)yi = ςiF (y1, . . . , yk, . . . , yn)z + ςiy
∗
kD(y1, . . . , α

−1(z), . . . , yn).

Rearranging by putting yi = yiz in the above equation, we get

F (ς1, . . . , ςk, . . . , ςn)yi − ςiF (y1, . . . , yk, . . . , yn)z = ςiy
∗
kD(y1, . . . , α

−1(z), . . . , yn),

for every ς1, . . . , ςk, . . . , ςn, y1, . . . , yk, . . . , yn ∈ R. Taking z as a common factor and
using (20), we have

ςiy
∗
kD(y1, . . . , α

−1(z), . . . , yn) = 0, for every ςi, y1, . . . , yk, . . . , yn ∈ R.

It follows that

ςiykD(y1, . . . , z, . . . , yn) = 0. for every ςi, y1, . . . , yk, . . . , yn ∈ R.

A simple manipulation yields that

D(y1, . . . , z, . . . , yn)ςiD(y1, . . . , z, . . . , yn)ykD(y1, . . . , z, . . . , yn)ςi = 0,

for every ςi, y1, .., yk, z ∈ R. Making use of semi-primeness of R, it follows that

D(y1, . . . , z, . . . , yn) = 0 for every y1, .., yi, z ∈ R.

Hence F acting as a left α-centralizer.

Acknowledgements

The authors are extremely grateful to the reviewers and editor for their generous sug-
gestions, insightful remarks and recommendations to make this manuscript well organized.

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



F. Shujat, S. Alharbi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6044 11 of 11

References

[1] A. Ali, V. De Filippis, and F. Shujat. On one sided ideals of a semiprime ring with
generalized derivations. Aequationes Mathematicae, 86(1-2):1–9, 2012.

[2] A. Z. Ansari, F. Shujat, A. Kamil, and A. Fallatah. Jordan φ-centralizers on
semiprime and involution rings. European Journal of Pure and Applied Mathematics,
18(1):5493–5508, 2025.

[3] S. Ali and M. S. Khan. On ∗-bimultipliers, generalized ∗-biderivations and related
mappings. Kyungpook Mathematical Journal, 51(3):301–309, 2011.

[4] A. Z. Ansari and F. Shujat. Jordan ∗-derivations on standard operator algebras.
Filomat, 37(1):37–41, 2023.

[5] N. Divinsky. On commuting automorphisms of rings. Transactions of the Royal
Society of Canada, Section III, 49:19–22, 1955.

[6] J. Luh. A note on commuting automorphisms of rings. The American Mathematical
Monthly, 77(1):61–62, 1970.

[7] B. Zalar. On multipliers of semiprime rings. Commentationes Mathematicae Univer-
sitatis Carolinae, 32(4):609–614, 1991.

[8] B. E. Johnson. An introduction to the theory of centralizers. Proceedings of the
London Mathematical Society, 14(3):299–320, 1964.

[9] A. N. Khan and S. Ali. Involution on prime rings with endomorphisms. AIMS
Mathematics, 5(4):3274–3283, 2020.

[10] R. Larsen. An Introduction to the Theory of Multipliers. Springer-Verlag, Berlin,
1971.

[11] J. K. Wang. Multipliers of commutative Banach algebras. Pacific Journal of Mathe-
matics, 11(3):1131–1149, 1961.

[12] M. Brešar and J. Vukman. On some additive mappings in rings with involution.
Aequationes Mathematicae, 38(2-3):178–185, 1989.

[13] J. H. Mayne. Centralizing automorphisms of prime rings. Canadian Mathematical
Bulletin, 19(1):113–115, 1976.

[14] I. N. Herstein. Rings with Involution. University of Chicago Press, Chicago, 1976.