©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 5doi: 10.28924/ada/ma.4.5
A New Study on Generalized Reverse Derivations of Semi-prime Ring

Muhammad Naeem Abbas1,∗, Mukhtar Ahmad1,∗, Abdul Rauf Khan2, Ather Qayyum3, Siti SuzlinSupadi3
1Department of Mathematics, Khawaja Fareed University of Engineering and Information Technology

Rahim Yar Khan, Pakistan
itxmemuktar@gmail.com, naeemabbas995@gmail.com

2Department of Mathematics, Ghazi University, D.G.Khan, Pakistan
arkhan@gudgk.edu.pk

3Institute of Mathematical Sciences, Universiti Malaya, Malaysia
dratherqayyum@um.edu.my, suzlin@um.edu.my

∗Correspondence: naeemabbas995@gmail.com, itxmemuktar@gmail.com

Abstract. The aim of this paper is to extend the ideas from Generalized reverse derivation to Gener-alized (α, β)-reverse derivations on Semi-prime ring. We prove that, if 0 6= d be reverse derivationin R and a Generalized (α, β)-reverse derivation g, then g is β-strong commutative preserved. Nextwe can prove that R is commutative.

1. Introduction
The study of centralizing mapping of semi-prime rings given by Bell and Martindale [3]. Belland Martindale [3] proved that [d(u1), u1]α,β = 0 ∀ u1 ∈ B, where 0 6= d a derivation of R and

R is semi-prime ring, then commutativity holds in R. Bell and Daif were studied the commu-tativity in prime and semi-prime rings that bind endomorphism or a derivation that preserves a
β-strong commutativity on a non-zero ideal right in [2]. Further, Ali and Shah [1] extend some con-sequences for generalized derivation of Bell and Martindale [3]. Bresar established that, if B 6= 0is left ideal in R a prime ring, and two mappings d1 and d2 are (α, β)-derivations in R satisfies
(d1α(a)− β(a)d2) ∈ Z(R), for each a ∈ B, so commutativity holds in R [5]. Some properties arestudied by Vukman in [12] and [4]. M. Samman and N. AL Yamani [8] studied reverse derivation onsemi prime rings. They proved that the mapping d : R → R is central derivation iff it is reversederivation and also that d 6= 0 a reverse derivation in semi-prime ring R, then the commutativityexists in R resently Mukhtar Ahmad et.al[9]. Later, the idea of revers derivation and some properties

Received: 19 Nov 2023.
Key words and phrases. Semi-prime ring; ideal; r-generalized reverse derivation; derivation; reverse derivation;r-generalized derivation and reverse derivation. 1

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https://doi.org/10.28924/ada/ma.4.5


Eur. J. Math. Anal. 10.28924/ada/ma.4.5 2of reverse derivation were studied by Bresar and Vukman [4]. The aim of this paper is extention thenotion of generalized reverse derivation to generalized (α, β)-reverse derivation resently MukhtarAhmad et.al[10]. A mapping G : R → R which associate with (α, β)-reverse derivation D is saidto be a generalized (α, β)-reverse derivation if, G(u1v1) = G(v1)α(u1) + β(v1)D(u1)resently R.M. Kashif et.al[11].
2. Preliminaries

Throughout this paper,
Definition 2.1. Let R is ring and it is considered as a semi-prime ring iff for any u1; u1 6= 0such that u1Ru1 = 0 implies u1 = 0.
Definition 2.2. The additive mapping d1 : R → R is known as (α, β)-derivation, if d1(u1v1) =

d1(u1)α(v1) + β(u1)d1(v1) hold ∀ u1, v1 ∈ R, where α and β are automorphism.
Definition 2.3. The mapping d1 : R → R is called a (α, β)-reverse derivation if d1(u1v1) =

d1(v1)α(u1) + β(v1)d1(u1) holds ∀ u1, v1 ∈ R, where α and β are automorphism.
Definition 2.4. An additive mapping H : R → R be a right (left) generalized (α, β)-reversederivation if there is a derivation d from R to R such thatH(u1v1) = H(v1)α(u1)+β(v1)d(u1) (H(u1v1) =

d(v1)α(u1) + β(v1)H(u1) for all u1, v1 ∈ R. H be a generalized reverse (α, β) of R associatedwith (α, β) derivation.
Definition 2.5. Some identities holds for every u1, v1, w1 ∈ R [u1, v1w1] = v1[u1, w1]+[u1, v1]w1

[u1v1, w1] = [u1, w1]v1 + u1[v1, w1]

[u1v1, w1]α,β = u1[v1, w1]α,β +[u1, β(w1)]v1 = u1[v1, α(w1)] + [u1, w1]α,β v1

[u1, v1w1]α,β = β(v1)[u1, w1]α,β +[u1, v1]α,β α(w1)

Definition 2.6. The derivation H would be commuting, if 0 = [v1, H(u1)], ∀ u1, v1 ∈ R.
Definition 2.7. The strong commutativity preserving is defined as [g(u1), g(v1)] = [u1, v1] forall u1, v1 ∈ R, where g : R→ R is a mapping on R.
Lemma 2.8. Let u1 6= 0 in Z(center of ring), if u1, v1 ∈ Z , then v1 ∈ Z.
Lemma 2.9. Let g : R → R be an additive map and on a left ideal B of R, g is centralizing,then g(u1) ∈ R ∀ u1 ∈ B ∪ Z.
Lemma 2.10. Let 0 6= B be an ideal of a semi-prime ring R. If the set [B,B] centralizes Z in

R, then B centralizes Z.
2.1. Point-Wise Operation. Theorem 2.11. Suppose 0 6= d from R to R a derivation in a semi-prime ring R. Let generalized (α, β)-reverse derivation g on a left ideal B 6= 0 of R. Then
g satisfies [g(w1), g(v1)] = β([w1, v1]) for all v1, w1 ∈ B (that is, g is β-strong commutativitypreserved), when g is a homomorphism on B.

Proof. Since g is generalized (α, β)-reverse derivation and homomorphism on B, such that
g(u1v1) = g(u1)g(v1)∀ u1, v1 ∈ B.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.5 3This implies
g(u1v1) = g(u1)g(v1) = g(v1)α(u1) + β(v1)d(u1), f or al l u1, v1 ∈ B. (1)
We replace v1 by v1w1 where w1 ∈ B, in equation (1), we obtain
g(u1)g(v1w1) = g(v1w1)α(u1) + β(v1w1)d(u1)this gives
g(u1)g(v1w1) = g(u1v1w1) = g(v1)g(w1)α(u1) + β(v1w1)d(u1),

f or al l u1, v1 ∈ B. (2)
As g is homomorphism, so we get
g(u1)g(v1w1) = g(u1)g(v1)g(w1) = g(u1v1)g(w1)this equalized to
g(u1v1)g(w1) = (g(v1)α(u1) + β(v1)d(u1))g(w1)this relates to
g(u1v1)g(w1) = g(v1)α(u1)g(w1) + β(v1)d(u1)g(w1)By the equation (2), we get
g(u1v1)g(w1) = g(v1)g(w1)α(u1) + β(v1)d(u1)g(w1), f or al l u1, v1 ∈ B. (3)
From equation (2) and equation (3), we obtain
β(v1)d(u1)g(w1) = β(v1)d(u1)β(w1)this implies
β(v1)d(u1)(g(w1)− β(w1)) = 0, f or al l u1, v1 ∈ B. (4)
Put w1 = [w1, v1] in equation (4), we have
β(v1)d(u1)(g([w1, v1])− β([w1, v1])) = 0,we arrives to
d(u1)β(v1)(g([w1, v1])− β([w1, v1])) = 0.By replacing β(v1) by (g([w1, v1])− β([w1, v1]))α(r)d(u1), we obtain
d(u1)(g([w1, v1])− β([w1, v1]))α(r)d(u1)(g([w1, v1])− β([w1, v1])) = 0,it gives
d(u1)(g([w1, v1])− β([w1, v1]))Rd(u1)(g([w1, v1])− β([w1, v1])) = 0.As R semi-prime, so we obtain
d(u1)(g([w1, v1])− β([w1, v1])) = 0,since d 6= 0, we have
g([w1, v1])− β([w1, v1]) = 0,we get
g([w1, v1]) = β([w1, v1])

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Eur. J. Math. Anal. 10.28924/ada/ma.4.5 4As g is homomorphism, so we have
[g(w1), g(v1)] = β([w1, v1])So g is β-strong commutative preserved on B.

Theorem 2.12. Let g on a left ideal B 6= 0 of R, is generalized (α, β)-reverse derivation. If gis homomorphism on B, then on B, g is commuting.
Proof. By theorem 2.11, g is β-strong commutative preserved, then ∀ u1, v1 ∈ B, we get

β([u1, v1]) = [g(u1), g(v1)] (5)
Replace v1 = v1u1 in equation (5) we have
β([u1, v1u1]) = [g(u1), g(v1u1)]

β([u1, v1])β(u1) = [g(u1), g(v1)]g(u1)By equation (5), we get
β([u1, v1])β(u1) = β([u1, v1])g(u1)this implies

β([u1, v1])(g(u1)− β(u1)) = 0. (6)
Now put v1 = u1v1 in equation (5) we have
β([u1, u1v1]) = [g(u1), g(u1v1)]

β(u1)β([u1, v1]) = g(u1)[g(u1), g(v1)]By equation (5), we have
β(u1)β([u1, v1]) = g(u1)β([u1, v1])this implies

(g(u1)− β(u1))β([u1, v1]) = 0. (7)
Put v1 = r1v1 in equation (6), we have
β([u1, r1v1])(g(u1)− β(u1)) = 0,this implies
β([u1, r1])β(v1)(g(u1)− β(u1)) = 0,we get
β([u1, r1])B(g(u1)− β(u1)) = 0,also that
β([u1, r1])RB(g(u1)− β(u1)) = 0,By semi-primeness of R, there exist a family w = {Pθ/θ ∈ ∧

} of prime ideals such that ⋂
Pθ =

0.If w has a member P and u1 ∈ B, then last relation, we get, B(g(u1) − β(u1)) not in P or
[β(u1), R] ⊆ P . If ∃ v1 ∈ B such that [β(u1), R] not in P . It implies B(g(v1) − β(v1)) ⊆ P . Let
w1 ∈ B is arbitrary such that [β(v1+w1), R] ⊆ P . This means that [β(w1), R] not in P and hence
(g(w1)− β(w1)) ⊆ P . In other ways [β(v1 +w1), R] ⊆ P , then B(g(v1 +w1)− β(v1 +w1)) ⊆ P .

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Eur. J. Math. Anal. 10.28924/ada/ma.4.5 5It gives B(g(w1)− β(w1)) ⊆ P .We obtain B(g(w1)−β(w1)) ⊆ P for every w1 ∈ B and hence [B,B](g(w1)−β(w1)) ⊆ P ∀ w1 ∈ B.As P is arbitrary and ⋂
Pθ = 0, this implies [B,B](g(w1)− β(w1)) = 0 for all w1 ∈ B. Similarly,we can show that (g(w1)− β(w1))[B,B] = 0 for all w1 ∈ B. This implies that (g(w1)− β(w1)) ∈

CR[B,B], for all w1 ∈ B. By Lemma 2.10 and [6], we have (g(u1), β(u1)) ∈ CR(B), ∀ w1 ∈ B.Thus we have [g(u1) − u1, β(u1)] = 0 ∀ u1 ∈ B. This implies that [g(u1), β(u1)] = 0 ∀ u1 ∈ B.This shows that g is commuting on B.
Theorem 2.13. Suppose a derivation, d : R → R where 0 6= d , in R and a generalized (α, β)-reverse derivation g on left ideal B 6= 0. If g is a homomorphism on B, then commutativity existsin R.
Proof. By our hypothesis

[g(u1), u1]α,β = 0, f or al l u1 ∈ B. (8)
We replace u1 by u1 + v1, in equation (2.1),we get
[g(u1 + v1), u1 + v1]α,β = 0,we have
[g(u1) + g(v1), u1 + v1]α,β = 0,we arrives to
[g(u1) + g(v1), u1]α,β +[g(u1) + g(v1), v1]α,β = 0,this gives
[g(u1), u1]α,β +[g(v1), u1]α,β +[g(u1), v1]α,β +[g(v1), v1]α,β = 0.By equation (), we obtain
[g(u1), v1]α,β +[g(v1), u1]α,β = 0, f or al l u1 ∈ B. (9)
By substituting v1 = u1v1 in equation (9), we have
[g(u1), u1v1]α,β +[g(u1v1), u1]α,β = 0,we have
β(u1)[g(u1), v1]α,β +[g(u1), u1]α,β α(v1) + [g(v1)α(u1) + β(v1)d(u1), u1]α,β = 0.this implies us by the equation (2.1),
β(u1)[g(u1), v1]α,β +[g(v1)α(u1), u1]α,β +[β(v1)d(u1), u1]α,β = 0.This gives us by [α(u1), α(u1)] = 0,
β(u1)[g(u1), v1]α,β +[g(v1), u1]α,β α(u1) + [β(v1)d(u1), u1]α,β = 0,Since g is commuting on B, we have
[β(v1)d(u1), u1]α,β = 0, f or al l u1 ∈ B. (10)
We replace v1 by r1v1 in equation (10), we have
[β(r1v1)d(u1), u1]α,β = 0,

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Eur. J. Math. Anal. 10.28924/ada/ma.4.5 6we get
β(r1)[β(v1)d(u1), u1]α,β +[β(r1), β(u1)]β(v1)d(u1) = 0.By equation (10), we have
[β(r1), β(u1)]β(v1)d(u1) = 0,this gives
[β(r1), β(u1)]Bd(u1) = 0, for all u1 ∈ B and r1 ∈ R,By the semi-primeness of R, ∃ a set ω = {Pα/α ∈ ∧

} of prime ideals and ⋂
Pα = (0).If P ∈ ω and u1 ∈ B, then by equation (10), [R, β(u1)] ⊆ P or P ⊇ d(u1). Since 0 6= d on R, soby [7], 0 6= d on B. Consider d(u1)P , where u1 ∈ B, then P ⊇ [R, β(u1)]. Suppose w1 ∈ B, wesee that w1 not in Z, then d(w1) ⊆ P and u1 + w1 not in Z. This gives that d(u1 + w1) ⊆ P andthen d(u1) ⊆ P , which contradicts to our consideration that d(u1)P . So, this gives us w1 ∈ Z, ∀

w1 ∈ B.This implies that B is commutative also that by the [7], then commutativity holds in R.
Theorem 2.14. Suppose a semi-prime ring R and a left ideal B of R, s.t. B⋂

Z 6= 0 for center
Z of R. Let a generalized (α, β)-reverse derivation g on R and d 6= 0 a derivation and g iscentralizing on B. Then commutativity holds in R.

Proof. If Z 6= 0 and g is commutation on B, so our proof is complete.As g is centralizing B and by Theorem 2.12, we get
[g(u1), u1]α,β ∈ Z, ∀ u1 ∈ B. (11)
Put u1 = (u1 + v1) in equation (11), then
[g(u1 + v1), u1 + v1]α,β ∈ Z, for all u1 ∈ B,this relates to
[g(u1), u1 + v1]α,β +[g(v1), u1 + v1]α,β ∈ Z, ∀ u1 ∈ B.It implies
β(u1)[g(u1), u1]α,β +β(u1)[g(u1), v1]α,β +[g(v1), u1]α,β α(u1) + [g(v1), v1]α,β α(u1) ∈ Z, ∀ u1 ∈
B.By the equation (11), we get
β(u1)[g(u1), v1]α,β +[g(v1), u1]α,β α(u1) ∈ Z, f or al l u1, v1 ∈ B. (12)
Replace u1 by v1w1 in equation (12), we obtain
β(u1)[g(v1w1), v1]α,β +[g(v1), v1w1]α,β α(u1) ∈ Z,we get
β(u1)[g(w1)α(v1) + β(w1)d(v1), v1]α,β +β(v1)[G(v1), w1]α,β α(u1)

+ [g(v1), v1]α(w1)α(u1) ∈ Z, this implies
β(u1)[g(w1)α(v1), v1]α,β +β(u1)[β(w1)d(v1), v1]α,β +β(v1)[g(v1), w1]α,β ∈ Z.This equalized to

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Eur. J. Math. Anal. 10.28924/ada/ma.4.5 7

β(u1)[g(w1), v1]α(v1) + β(u1)g(w1)[α(v1), α(v1)]α,β +β(u1)[β(w1), β(v1)]d(v1)

+ β(u1)β(w1)[d(v1), v1]α,β +β(v1)[g(v1), w1]α,β α(u1) ∈ Z.As we know for any u1, v1, w1 ∈ Z can commute with each one of R, by equation (12) and
[α(w1), α(v1)] = 0, we get
β(u1)β(w1)[d(v1), v1]α,β ∈ Z.Since β(w1) 6= 0, this implies by Lemma 2.8, we get
[d(v1), v1]α,β ∈ Z, for every v1 ∈ B.Then we have d is centralizing on B, hence by the reference [3], R is commutative. This completesour proof.

References
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https://doi.org/10.28924/ada/ma.4.5

	1. Introduction
	2. Preliminaries
	2.1. Point-Wise Operation

	References