# Centrally Extendedα-Homoderivations on Prime and Semiprime Rings

Authors: Mahmoud M. El-Soufi; A. Ghareeb
Journal: Journal of Mathematics
 (2022)

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## Abstract

We present a new type of mappings called centrally extendedα-homoderivations of a ring ℜ (i.e., a map H from ℜ into ℜ which satisfies Hx+y−Hx−Hy∈Zℜ and Hxy−HxHy−Hxαy−αxHy∈Zℜ for any x,y∈ℜ) where α is a mapping of ℜ and discuss the relationship between these mappings and other related mappings. Also, we study their effect on the center of a ring, and we give several outcomes on the commutativity of the ring.


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## Body

## 1. Introduction

All through this paper,Zℜ will indicate the center of an associative ring ℜ, and α and β are mappings of ℜ. If xℜy=0 implies either x=0 or y=0, a ring ℜ is said to be prime. A ring ℜ is known to be semiprime if the zero ideal is a semiprime ideal (or if xℜx=0 implies x=0). The x,y symbol is utilized to mean the xy−yx and x∘y symbol would signify xy+yx. An additive mapping D of ℜ is said to be a derivation if Dxy=Dxy+xDy holds for all x,y∈ℜ. Following [1–3], an additive mapping D of ℜ is called an α,β-derivation if Dxy=Dxαy+βxDy holds for all x,y∈ℜ.The definition of centrally extendedCE− derivation of a ring ℜ has been introduced in a recent paper by Bell and Daif [4]. A centrally extended derivation of a ring ℜ is a map D of ℜ such that Dx+y−Dx−Dy∈Zℜ and Dxy−Dxy−xDy∈Zℜ for each x,y∈ℜ. In [5], Tammam et al. have introduced the concept of CEα,β derivations of ℜ. A CEα,β derivation of ℜ is a map D of ℜ such that Dx+y−Dx−Dy and Dxy−Dxαy−βxDy∈ZR for each x,y∈ℜ.In 2000, El-Soufi [6] has introduced the concept of a homoderivation of a ring ℜ. An additive map H of a ring R is called a homoderivation if Hxy=HxHy+Hxy+xHy,∀x,y∈ℜ.In the current paper, we present the concept of aCEα-homoderivation, and we obtain some results concerning CEα-homoderivations and commutativity of rings.Definition 1.
Letℜ be a ring.(i)
A centrally extendedCE-homoderivation is a map H of ℜ which fulfills(1)Hx+y−Hx−Hy∈Zℜ,∀x,y∈ℜ,Hxy−HxHy−Hxy−xHy∈Zℜ,∀x,y∈ℜ.(ii)
Anα-homoderivation is an additive map H of ℜ, and it achieves(2)Hxy=HxHy+Hxαy+αxHy,∀x,y∈ℜ.(iii)
ACEα-homoderivation is a mapping H from ℜ into ℜ which satisfies(3)Hx+y−Hx−Hy∈Zℜ,∀x,y∈ℜ,Hxy−HxHy−Hxαy−αxHy∈Zℜ,∀x,y∈ℜ.
In 1985, Chung [7] has introduced the notion of nil and nilpotent derivations: Let ℜ be a ring and δ a derivation of ℜ. Then, δ is said to be nil if, for any x∈ℜ, there is a +ve integer m=mx such that δmx=0. Here, if the number m can be taken freely of x, the derivation δ is called nilpotent.Definition 2.
LetU be a subset of a ring ℜ and H a map of ℜ. H is called nilpotent on U if there exists an integer m>1 such that Hmx=0,∀x∈U.Definition 3.
LetU be a subset of a ring ℜ. The two maps α,H of ℜ commute on U if αH=Hα on U.Remark 1.
A homoderivation ofℜ is an Iid-homoderivation, and a CE-homoderivation of ℜ is a CEIid-homoderivation of ℜ, where Iid is the identity map of ℜ. Any map commutes with Iid.


## 2. Examples Indicating the Existence ofCEα-Homoderivations

In the following example, we make sure there is anα-homoderivation map.Example 1.
Letℜ=ℤ6. We define the two maps α,H:ℜ⟶ℜ where αx=4x and Hx=2x,∀x∈ℜ. Here, α is an endomorphism, H is an α-homoderivation map, and α commutes with H.
It is clear by Definition1 that every α-homoderivation is a CEα-homoderivation, but the converse (in general) is not correct by the following examples.Example 2.
Letℜ=ℤ6. We define the map H:ℜ⟶ℜ where Hx=2−x,∀x∈ℜ. Thus, H is a CE-homoderivation but not a homoderivation, since for any x,y∈ℜ, we have Hx+y−Hx−Hy=4, a nonzero central element and also Hxy−HxHy−Hxy−xHy=4, also a nonzero central element.Example 3.
Letℜ=ℤ6. Hence, ℜ=Zℜ. We define the two maps α,H:ℜ⟶ℜ such that αx=4x and Hx=2x+x2,∀x∈ℜ. Here, α is an endomorphism and H is a CEα-homoderivation but not an α-homoderivation map, since for all x,y∈ℜ, we have Hxy−HxHy−Hxαy−αxHy∈Zℜ but Hx+y−Hx−Hy=2xy∈Zℜ (where ℤ6 is a commutative ring) and 2xy≠0,∀x,y∈ℜ.Example 4.
Consider the infinite ringℜ=ℤ6⊕M2ℤ. We define the two maps α,H:ℜ⟶ℜ such that αx,A=4x,A and Hx,A=2−x,−A,∀x,A∈ℜ. Suppose that x,A,u,B∈ℜ, then we have αx,A+u,B=αx+u,A+B=4x+4u,A+B. On other hand, αx,A+αu,B=4x,A+4u,B=4x+4u,A+B. So αx,A+u,B=αx,A+αu,B. Also, we have αdx,Au,B=αxu,AB=4xu,AB. On other hand, αx,Aαu,B=4x,A4u,B=4xu,AB. So, αx,Au,B=αx,Aαu,B. Moreover, Hx,A+u,B−Hx,A−Hu,B=4,0∈Zℜ and Hx,Au,B−Hx,AHu,B−Hx,Aαu,B−αx,AHu,B=4,0 ∈Zℜ. Therefore, α is an endomorphism, and H is a CEα-homoderivation but not an α-homoderivation map.
In the following example, we make sure there is a nilpotent map on a ringℜ, and we illustrate the influential role of the map α.Example 5.
Letℜ be a ring such that charℜ=4. The map α:ℜ×ℜ⟶ℜ×ℜ where αa,b=b,a,∀a,b∈ℜ×ℜ is an endomorphism. The map H:ℜ×ℜ⟶ℜ×ℜ where Ha,b=b−a,a−b,∀a,b∈ℜ×ℜ commutes with α, nilpotent (where H3x=0,∀x∈ℜ×ℜ) and is an Iid-homoderivation on ℜ but not an α-homoderivation on ℜ.


## 3. Rings withCEα-Homoderivations

By Definition1, every α-homoderivation is a CEα-homoderivation but the opposite (in general) is not correct. In this section, we see when a CEα-homoderivation is an α-homoderivation. The next results are dependent on the absence of a nonzero central ideal in the rings. Also, we give some properties of CEα-homoderivations.Note that for two elementsx and y of ℜ, zHx,y,+ means the central element resulting from the effect of the map H on x+y. Likewise, zHx,y,. means the central element resulting from the effect of the map H on x.y.Theorem 1.
Letα be an epimorphism of a ring ℜ and H a nilpotent CEα-homoderivation of ℜ which commutes with α on ℜ. If the only central ideal of ℜ is zero ideal, then H is additive on ℜ.Proof.
For any fixedx,y∈ℜ, we have(4)Hx+y=Hx+Hy+zHx,y,+.
For arbitraryω∈ℜ, we have(5)Hx+yω=Hx+yHω+Hx+yαω+αx+yHω+zHx+y,ω,.=Hx+Hy+zHx,y,+Hω+αω+αxHω+αyHω+zHx+y,ω,..
Then again, we have(6)Hx+yω=Hxω+yω=Hxω+Hyω+zHxω,yω,+=HxHω+Hxαω+αxHω+HxHω+Hyαω+αyHω+zHxω,yω,++zHx,ω,.+zHy,ω,..
Comparing (5) and (6), we get(7)Hω+αωzHx,y,+∈Zℜ,∀ω∈ℜ.
SinceH is nilpotent on ℜ, there exists an integer n>1 such that Hnx=0,∀x∈ℜ. Replacing ω by Hn−1ω in (7), using α and H commute, and H is nilpotent, we get(8)αHn−1ωzHx,y,+∈Zℜ,∀ω∈ℜ.
Sinceα is an epimorphism, then(9)Hn−1ωzHx,y,+∈Zℜ,∀ω∈ℜ.
Replacingω by Hn−2ω in (7), we get(10)Hn−1ω+αHn−2ωzHx,y,+∈Zℜ,∀ω∈ℜ.
Again, by (9) and α is onto, we obtain(11)Hn−2ωzHx,y,+∈Zℜ,∀ω∈ℜ.
So, we can repeat the above argument to get(12)HωzHx,y,+∈Zℜ,∀ω∈ℜ.
By (7) and (12), we get αωzHx,y,+∈Zℜ,∀ω∈ℜ. But α is an epimorphism, so ωzHx,y,+∈Zℜ,∀ω∈ℜ. Therefore, ℜzHx,y,+⊂Zℜ. Thus, ℜzHx,y,+=0. Let Aℜ be the two-sided annihilator of ℜ, we have zHx,y,+∈Aℜ. Since Aℜ is a central ideal, therefore, zHx,y,+=0, and by (4), Hx+y=Hx+Hy.Theorem 2.
Letα be an epimorphism of a semiprime ring ℜ. If the only central ideal of ℜ is zero ideal, then every nilpotent CEα-homoderivation of ℜ which commutes with α is an α-homoderivation of ℜ.Proof.
LetH be a nilpotent CEα-homoderivation which commutes with α on ℜ. According to Theorem 1, H is additive. Let x,y,ω∈ℜ be arbitrary elements. Then,(13)Hxy=HxHy+Hxαy+αxHy+zHx,y,.,(14)Hyω=HyHω+Hyαω+αyHω+zHy,ω,..
By using the associativity ofR, we have Hxyω=Hxyω, so(15)zHx,y,.Hω+αω−zHy,ω,.Hx+αx∈Zℜ.
Therefore,(16)zHx,y,.Hω+αω,Hx+αx=0,∀x,y,ω∈ℜ.
Replacingω by Hn−1ω in (16), we have(17)zHx,y,.αHn−1ω,Hx+αx=0,∀x,y,ω∈ℜ.
Sinceα and H commute and α is onto, then(18)zHx,y,.Hn−1ω,Hx+αx=0,∀x,y,ω∈ℜ.
Replacingω by Hn−2ω in (16) and using (18), we have(19)zHx,y,.Hn−2ω,Hx+αx=0,∀x,y,ω∈ℜ.
Repeating the step above, we get(20)zHx,y,.Hω,Hx+αx=0,∀x,y,ω∈ℜ.
From (16) and (20), we obtain(21)zHx,y,.αω,Hx+αx=0,∀x,y,ω∈ℜ.
Again, sinceα is onto, then(22)zHx,y,.ω,Hx+αx=0,∀x,y,ω∈ℜ.
Replacingω by ωb in (22), we get(23)zHx,y,.ωb,Hx+αx=0,∀x,y,ω,r∈ℜ.
Therefore,(24)zHx,y,.ℜb,Hx+αx=0,∀x,y,r∈ℜ.
LetΓ=Qμ|μ∈Λ be a set of prime ideals of ℜ and ∩Qμ=0. Also, let Q indicates a typical Qμ in Γ. For each x∈ℜ, either zHx,y,.∈Q,∀y∈ℜ or b,Hx+αx∈Q,∀b∈ℜ. In the first case, zHx,y,.+Q=Q,∀y∈ℜ. Thus, ℜzHx,y,.+Q=Q,∀y∈ℜ. So, ℜzHx,y,.+Qu+Q=u+QℜzHx,y,.+Q,∀y,u∈ℜ. Therefore, ℜzHx,y,.,u+Q=Q,∀y,u∈ℜ. Thus, ℜzHx,y,.,u∈∩Qμ=0,∀y,u∈ℜ. That is, ℜzHx,y,.⊂Zℜ,∀y∈ℜ. Let Aℜ be the two-sided annihilator of ℜ, we have zHx,y,+∈Aℜ. Since Aℜ is a central ideal, therefore, zHx,y,.=0,∀y∈ℜ. In the second case, b,Hx+αx∈Q,∀b∈ℜ. Thus, b,Hx+αx+Q=Q,∀b∈ℜ implies that(25)b+Q,Hx+αx+Q=Q,∀b∈ℜ.
From (15) and (25), we have(26)Q=zHx,y,.Hω+αω+Q,−zHy,ω,.Hx+αx+Q,b+Q=zHx,y,.Hω+αω+Q,b+Q,∀y,ω,b∈ℜ.
As above in equation (16), we arrive at Q=zHx,y,.ω+Q,b+Q=zHx,y,.ω,b+Q,∀y,ω,b∈ℜ. Therefore, zHx,y,.ω,b∈Q,∀y,ω,b∈ℜ. So, zHx,y,.ω,b∈∩Qμ=0,∀y,ω,b∈ℜ. Again, zHx,y,.=0,∀y∈ℜ. So, we have zHx,y,.=0,∀x,y∈ℜ. From (13), we have(27)Hxy=HxHy+Hxαy+αxHy.
Thus,H is an α-homoderivation of ℜ.Corollary 1.
Letℜ be a semiprime ring. If the only central ideal of ℜ is zero ideal, then every nilpotent CE-homoderivation of ℜ is a homoderivation of ℜ.
At this stage, we present the answer to the following inquiry: When does aCEα-homoderivation preserves the center?Theorem 3.
Letα be an epimorphism of a ring ℜ. If Zℜ has no nonzero nilpotent elements, then any nilpotent CEα-homoderivation which commutes with α on ℜ preserves Zℜ.Proof.
Suppose thatH is a nilpotent CEα-homoderivation on ℜ which commutes with α, x∈ℜ and z∈Zℜ. Therefore,(28)Hzx−HzHx−Hzαx−αzHx∈Zℜ,(29)Hxz−HxHz−Hxαz−αxHz∈Zℜ.
Subtracting the two equations (28) and (29), we get(30)Hx+αx,Hz∈Zℜ,∀x∈ℜ.
PuttingHn−1x instead of x in (30) and since α is onto, α and H commute, we obtain(31)Hn−1x,Hz∈Zℜ,∀x∈ℜ.
Again, puttingHn−2x instead of x in (30) and using (31), we get αHn−2x,Hz∈Zx,∀x∈ℜ. Now, by properties of α and H, we have(32)Hn−2x,Hz∈Zℜ,∀x∈ℜ.
Repeating the same technique as above, we get(33)Hx,Hz∈Zℜ,∀x∈ℜ.
From (30) and (33), we have αx,Hz∈Zℜ,∀x∈ℜ. Since α is onto, then(34)x,Hz∈Zℜ,∀x∈ℜ.
Replacingx by xHz in (34) yields(35)xHz,Hz=x,HzHz∈Zℜ,∀x∈ℜ.
So, we havex,HzHz,x=0,∀x∈ℜ. Therefore,(36)x,Hz2=0,∀x∈ℜ.
But, the centerZℜ contains no nonzero nilpotent elements, so from (34) and (36) we obtain x,Hz=0,∀x∈ℜ. Hence, Hz∈Zℜ, that is, H preserves the center.Corollary 2.
Letα be an epimorphism of a semiprime ring ℜ. If H is a nilpotent CEα-homoderivation on ℜ which commutes with α, then H preserves Zℜ.Corollary 3.
IfZℜ has no nonzero nilpotent elements, then every nilpotent CE-homoderivation of ℜ preserves the center.


## 4. Commutativity of Prime Rings withCEα-Homoderivations

Throughout this section, we discuss the conditions that guarantee the commutativity of prime rings withCEα-homoderivation.Theorem 4.
Letα be an epimorphism of a prime ring ℜ and let H:ℜ⟶ℜ be a nilpotent CEα-homoderivation commute with α which is not α-homoderivation on ℜ. Then, ℜ is commutative.Proof.
Ifℜ has no nonzero central ideals, then by Theorem 2 we have H is an α-homoderivation on ℜ, which is a contradiction. Subsequently, ℜ has a nonzero ideal I and I⊂Zℜ. Thus, by [[8], Lemma 1(b)], ℜ is commutative.Corollary 4.
If a prime ringℜ has a nilpotent CE-homoderivation which is not homoderivation, then ℜ is commutative.Theorem 5.
Letα be an epimorphism of a prime ring ℜ and H:ℜ⟶ℜ a nilpotent CEα-homoderivation which commute with α. If H0 is a nonzero element, then ℜ is commutative.Proof.
We haveH0+0−H0−H0∈Zℜ because H is a CE-homoderivation, so H0∈Zℜ. Now, for all x∈ℜ, we have H0x−H0Hx−H0αx−α0Hx∈Zℜ, which implies that H0Hx+αx∈Zℜ. But H commutes with α and nilpotent on ℜ, and α is onto, we conclude that H0x∈Zℜ,∀x∈ℜ. Thus, H0x,r=0,∀r,x∈ℜ. By using H0∈Zℜ, we obtain H0x,r=0,∀x,r∈ℜ. Replacing x by yx, we get H0yx,r=0,∀x,y,r∈ℜ. Therefore, H0ℜx,r=0,∀x,r∈ℜ. By primeness of ℜ and H0≠0, x,r=0,∀x,r∈ℜ. Thus, ℜ is commutative.Corollary 5.
If a prime ringℜ has a nilpotent CE-homoderivation H such that H0 is a nonzero element, then ℜ is commutative.Theorem 6.
Letα be an epimorphism of a prime ring ℜ. If H is a nilpotent CEα-homoderivation which commutes with α on ℜ such that Hx,y=0 (or Hx∘y=0), ∀x,y∈ℜ, then H=0 or ℜ is commutative.Proof.
If there exists a nonzero central ideal inℜ, according to [[8], Lemma 1(b)] we have ℜ is commutative. Suppose that ℜ has no nonzero central ideals, by Theorem 1, we have H is additive. Let Hx,y=0,∀x,y∈ℜ. Substituting yx for x, we get Hyx,y=0=Hyx,y,∀x,y∈ℜ. Since H is a CEα-homoderivation of ℜ, then Hyαx,y∈Zℜ,∀x,y∈ℜ. But α is an epimorphism, then(37)Hxy,αx∈Zℜ,∀x,y∈ℜ.
Puttingxαy instead of x in (37), we obtain Hxx,αyαy∈Zℜ,∀x,y∈ℜ. Therefore,(38)r,Hyy,αyαy=0,∀x,y,r∈ℜ,which implies(39)Hyx,αyr,αy=0,∀x,y,r∈ℜ.
Puttingrs instead of r in (39) and using (39), we get(40)Hyx,αyrs,αy=0,∀x,y,r,s∈ℜ.
By hypothesis,ℜ is prime, so for each y∈ℜ either Hyx,αy=0,∀x∈ℜ or αy∈Zℜ. Suppose that y∈ℜ such that Hyx,αy=0,∀x∈ℜ. Replacing x by sx, we obtain Hysx,αy=0,∀x,s∈ℜ. Therefore, for each y∈ℜ, we have x,αy=0,∀x∈ℜ or Hy=0. Thus, for each y∈ℜ, either αy∈Zℜ or Hy=0. Assume that(41)A=y∈ℜ:Hy=0,B=y∈ℜ:αy∈Zℜ.
Then,A and B are additive subgroups and A∪B=ℜ. But ℜ,+ cannot be the union of two of its proper subgroups, then either A=ℜ implies H=0 or B=R implies ℜ is commutative, where α is onto. On other hand, suppose that Hx∘y=0,∀x,y∈ℜ. Substituting yx for x in Hx∘y=0, then we obtain Hyx∘y=Hyx∘y=0,∀x,y∈ℜ. Since H is a CEα-homoderivation of ℜ, then(42)Hyαx∘y∈Zℜ,∀x,y∈ℜ.
Substitutingxy for x in (42), we obtain(43)Hyαx∘yαy∈Zℜ,∀x,y∈ℜ.
By Lemma 4 in [9] for each y∈ℜ, we have either Hyαx∘y=0,∀x∈ℜ or αy∈Zℜ. Suppose that y∈ℜ such that Hyαx∘y=0,∀x∈ℜ. Replacing x by sx, we obtain Hyαsαx,y=0,∀x,s∈ℜ. Since α is an epimorphism, Hysx,αy=0,∀x,s∈ℜ. Since ℜ is prime, αy∈Zℜ or Hy=0. Therefore, for each y∈ℜ, there are two cases either Hy=0 or αy∈Zℜ. Thus, we have H=0 or ℜ is commutative.Corollary 6.
IfH is a nonzero nilpotent CE-homoderivation of a prime ring ℜ and Hx,y=0 (or Hx∘y=0) ∀x,y∈ℜ, then ℜ is commutative.



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