Research article

Sensitivity analysis of a non-Markovian feedback retrial queue, reneging, delayed repair with working vacation subject to server breakdown

  • Received: 15 April 2024 Revised: 25 May 2024 Accepted: 03 June 2024 Published: 28 June 2024
  • MSC : 60K25, 68M20, 90B22

  • This study investigated the steady-state characteristics of a non-Markovian feedback retrial queue with reneging, delayed repair, and working vacation. In this scenario, we assumed that consumers arrive through Poisson processes and the server provides service to consumers during both regular and working vacation periods. However, it is subject to breakdowns at any moment, resulting in a service interruption for a random duration. Additionally, the concept of delay time was also presented. The consumer that is dissatisfied with the service may re-enter the orbit to receive another service; this individual is considered a feedback consumer. The server will go on a working vacation if the orbit is empty after successfully serving a satisfied consumer. By utilizing the supplementary variable technique (SVT), we examined the steady-state probability generating function of the system and orbit sizes. Finally, numerical outcomes and a sensitivity analysis were given to verify the analytical findings of important performance indicators.

    Citation: S. Sundarapandiyan, S. Nandhini. Sensitivity analysis of a non-Markovian feedback retrial queue, reneging, delayed repair with working vacation subject to server breakdown[J]. AIMS Mathematics, 2024, 9(8): 21025-21052. doi: 10.3934/math.20241022

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  • This study investigated the steady-state characteristics of a non-Markovian feedback retrial queue with reneging, delayed repair, and working vacation. In this scenario, we assumed that consumers arrive through Poisson processes and the server provides service to consumers during both regular and working vacation periods. However, it is subject to breakdowns at any moment, resulting in a service interruption for a random duration. Additionally, the concept of delay time was also presented. The consumer that is dissatisfied with the service may re-enter the orbit to receive another service; this individual is considered a feedback consumer. The server will go on a working vacation if the orbit is empty after successfully serving a satisfied consumer. By utilizing the supplementary variable technique (SVT), we examined the steady-state probability generating function of the system and orbit sizes. Finally, numerical outcomes and a sensitivity analysis were given to verify the analytical findings of important performance indicators.


    In this work, Λ refers to an associative ring, and Z(Λ) denotes the center of Λ. The symbols [r,s] and rs stand for rssr and rs+sr, respectively. If SΛ, define F:ΛΛ to be centralizing on S if [s,F(s)]Z(Λ) for all sS; and is commuting on S if [s,F(s)]=0 for all sS. A mapping F is said to be strong commutativity-preserving on S if [s,t]=[F(s),F(t)] for all s,tS.

    Λ is considered prime if rΛs={0}, where r and s are both in Λ, implying that either r is zero or s is zero. This prime ring definition is equivalent to: (ⅰ) The product of two non-zero two-sided ideals of Λ is not zero. (ⅱ) The left annihilator of a non-zero left ideal is zero; for further information, see [1], page 47. Λ is considered semiprime if rΛr={0}, then r=0. If D(rs)=D(r)s+rD(s) holds for every r,s in Λ, then the additive map D is said to be a derivation.

    In a recent publication [2], Bell and Daif introduced the idea of a ring's centrally extended derivation (CE-derivation). Suppose that D is a mapping of a ring Λ. If D(s+u)D(s)D(u)Z(Λ) and D(su)D(s)usD(u)Z(Λ) for every s,uΛ, then D is known as a CE-derivation. The CE-(ρ,σ)-derivation on Λ has been described by Tammam et al. [3] as a map D on Λ achieves, for each s,uΛ, both

    D(s+u)D(s)D(u)andD(su)D(s)ρ(u)σ(s)D(u)areinZ(Λ).

    The concept of homoderivations of rings was first introduced by El-Soufi in 2000 [4]. A mapping on a ring Λ is defined as a homoderivation if it satisfies the relation (su)=s(u)+(s)u+(s)(u) for all s,uΛ, provided that is additive.

    The following are a few instances of homoderivations:

    Example 1.1. [4] Let Λ be a ring and ϕ be an endomorphism of Λ. Then, the mapping :ΛΛ defined by (u)=ϕ(u)u is a homoderivation of Λ.

    Example 1.2. [4] Let Λ be a ring. Then, the additive mapping :ΛΛ defined by (u)=u is a homoderivation of Λ.

    Example 1.3. [4] Let Λ=Z(2), a ring of all the real numbers of the form u+s2 such that u,sZ, the set of all the integers, under the usual addition and multiplication of real numbers. Then, the map :ΛΛ defined by (u+s2)=2s2 is a homoderivation of Λ.

    Melaibari et al. [5] demonstrated the commutativity of a prime ring Λ in 2016 by admitting a non-zero homomoderivation that satisfies any one of the following requirements: ⅰ. [v,u]=[(v),(u)], for all v,uU, non-zero ideal of Λ; ⅱ. ([v,u])=0, for all v,uU, a non-zero ideal of Λ; or ⅲ. ([v,u])(Λ), for all v,uΛ.

    According to Alharfie et al. [6], a prime ring Λ is commutative if any of the following requirements are met: For all v,uI, ⅰ. v(u)±vuZ(Λ), ⅱ. v(u)±uvZ(Λ), or ⅲ. v(u)±[v,u]Z(R). is a homoderivation of Λ, and I is a non-zero left ideal of Λ.

    The commutativity of a semiprime (prime) ring admitting a homoderivation meeting certain identities on a ring was investigated in 2019 by Alharfie et al. [7] and Rehman et al. [8].

    Over the past few years, researchers [9,10,11] have obtained many significant results pertaining to different aspects of homoderivations.

    In 2022, Tammam et al. [12] extended the concept of homoderivations by introducing the notion of n-homoderivations, where n is an integer. A map n is known as an n-homoderivation if it fulfills the requirement n(su)=sn(u)+n(s)u+nn(s)n(u) for all s,uΛ, provided n is additive.

    We draw inspiration from Bell and Daif's study [2], building on the new concept of n-homoderivations introduced in [12]. We focus on exploring the notion of a centrally extended n-homoderivation (CE-n-homoderivation), where nZ, as an extension of the traditional definition of homoderivations. Moreover, we explore several results regarding the ring commutativity of a ring equipped with a CE-n-homoderivation fulfilling specific conditions.

    Definition 1.1. Let s and u be any two elements in Λ and n be an integer, and let Hn be a mapping on a ring Λ. If Hn achieves

    Hn(s+u)Hn(s)Hn(u)Z(Λ),andHn(su)Hn(s)usHn(u)nHn(s)Hn(u)Z(Λ),

    then Hn is called a CE-n-homoderivation.

    It is clear that the previous definition generalizes the idea of centrally extended derivations (CE-derivations) presented by Bell and Daif [2] to the general case of centrally extended homoderivations of the type n (CE-n-homoderivations).

    Chung was the first to develop the idea of nil and nilpotent derivations in [13]. Consider a ring Λ that has a derivation δ. δ is considered to be nil if k=k(r)Z+ occurs for every rΛ with δk(r)=0. If the integer k can be freely taken out of r, then the derivation δ is said to be nilpotent.

    Definition 1.2. Assume that SΛ and that H and ϕ are two maps on a ring Λ. For some kZ+{1}, H is considered nilpotent on S if Hk(S)={0}. If ϕ(H(s))=H(ϕ(s)), for every sS, then two mappings H and ϕ are said to be commute on S.

    Remark 1.1. According to our definition of a CE-n-homoderivation, we assert that

    (1) Any CE-0-homoderivation of Λ is a CE-derivation on Λ.

    (2) Any CE-1-homoderivation of Λ is a CE-homoderivation on Λ.

    (3) Any n-homoderivation is a CE-n-homoderivation, but the inverse (in general) is not true.

    Remark 1.2. θHn(r,s,+) and θHn(r,s,) refer to the central elements generated through the influence of Hn on the sum r+s and the product rs, respectively, for any two elements r,sΛ.

    Theorem 1.1. Given a ring Λ, let n be any arbitrary non-zero integer. If the following centrally additive map γn:ΛΛ satisfies

    γn(st)=γn(s)t+sγn(t)+nγn(s)γn(t)+θHn(s,t,), (1.1)

    for each s,tΛ,θHn(s,t,)Z(Λ), then there exists a centrally extended homomorphism ϕn:ΛΛ such that ϕn(s)=s+nγn(s) for each sΛ.

    Proof. Clearly, since γn is a centrally additive, ϕn is centrally additive. Multiplying (1.1) with n leads to

    nγn(st)=nγn(s)t+nsγn(t)+nγn(s)nγn(t)+nθHn(s,t,) for all s,tΛ.

    If we add st to both sides of this equation, then

    nγn(st)+st=nγn(s)t+nsγn(t)+nγn(s)nγn(t)+st+nθHn(s,t,),

    for all s,tΛ. Observe however that

    nγn(s)t+nsγn(t)+nγn(s)nγn(t)+st=(nγn(s)+s)(nγn(t)+t),

    for all s,tΛ; revealing precisely that the mapping ϕn:ΛΛ specified by ϕn(s)=nγn(s)+s for all s,tΛ is a centrally extended homomorphism.

    Few adoptions on the proof of [12] Lemma 1 asserts that

    Lemma 1.1. Let K be a non-zero left ideal and Λ be a semi-prime ring. Hn is commuting on K if it is a centralizing CE-n-homoderivation on K.

    In this section, we confirm the presence of CE-n-homoderviation maps in the instances listed below.

    Example 2.1. Let Λ=M2(Z), the ring of 2×2 integer matrices, and let K be a nonzero central ideal of Λ. Suppose that fn:ΛK is any additive map and n:ΛΛ is any n-homoderivation of Λ. Therefore, the map Hn:ΛΛ such that Hn(x)=n(x)+fn(x), for all xΛ, is a CE n-homoderivation but it is not n-homoderivation.

    Example 2.2. Let Λ1 be a commutative domain, Λ2 a noncommutative prime ring with an n-homoderivation n, and Λ=Λ1Λ2. Define Hn:ΛΛ by Hn((s,u))=(g(s),n(u)), where g:Λ1Λ1 is a map that is not an n-homoderivation. Then, Λ is a semiprime ring, and Hn is a CE-n-homoderivation that is not an n-homoderivation. Furthermore, Λ1{0} is an ideal that is contained in the center of Λ.

    In this section, we explore the conditions under which a CE-n-homoderivation fulfills the requirements of an n-homoderivation. Additionally, it delves into the fundamental properties of CE-n-homoderivations.

    Throughout, Hn is a centrally extended n-homoderivation of a ring Λ, and nZ, ϕn will be the related CE-homomorphism to Hn defined in Theorem 1.1.

    Theorem 3.1. Let Λ be any ring containing no non-zero ideals in its center. Then, each nilpotent CE-n-homoderivation Hn on Λ is additive. Also, every CE-n-homoderivation Hn on Λ related to an epimorphism ϕn is additive.

    Proof. (i) If Hn is nilpotent:

    Let s,uΛ be two fixed elements. By assumption,

    Hn(s+u)=Hn(s)+Hn(u)+θHn(s,u,+). (3.1)

    So, for each vΛ, we obtain

    Hn((s+u)v)=(s+u)Hn(v)+Hn(s+u)v+nHn(s+u)Hn(v)+θHn(s+u,v,)=(Hn(s)+Hn(u)+θHn(s,u,+))(v+nHn(v))+uHn(v)+sHn(v)+θHn(s+u,v,). (3.2)

    However, we also have

    Hn((s+u)v)=Hn(sv+uv)=Hn(sv)+Hn(uv)+θHn(sv,uv,+)=Hn(s)v+sHn(v)+nHn(s)Hn(v)+uHn(v)+Hn(u)v+nHn(u)Hn(v)+θHn(sv,uv,+)+θHn(s,v,)+θHn(u,v,). (3.3)

    Comparing (3.2) and (3.3), we get

    (v+nHn(v))θHn(s,u,+)Z(Λ),for allvΛ. (3.4)

    Due to the fact that Hn is nilpotent, kZ,k>1 so that Hkn(s)=0 for all sΛ. By putting Hk1n(v) instead of v in (3.4), the result is

    Hk1n(v)θHn(s,u,+)Z(Λ), for each vΛ. (3.5)

    Putting Hk2n(v) instead of v in (3.4), we get

    (Hk2n(v)+nHk1n(v))θHn(s,u,+)Z(Λ), for each vΛ. (3.6)

    Once more, using (3.5), we get

    Hk2n(v)θHn(s,u,+)Z(Λ), for each vΛ. (3.7)

    Hence, we may repeat the preceding procedure to achieve

    Hn(v)θHn(s,u,+)Z(Λ), for each vΛ. (3.8)

    Using (3.4) and (3.8), we get vθHn(s,u,+)Z(Λ),for allvΛ. Thus, vθHn(s,u,+)Z(Λ),for allvΛ. Therefore, ΛθHn(s,u,+)Z(Λ). Thus, ΛθHn(s,u,+)={0}. If Ann(Λ) is the 2-sided annihilator of Λ, then θHn(s,u,+)Ann(Λ). However, Ann(Λ) is an ideal on Λ contained in Z(Λ), so θHn(s,u,+)=0. Therefore, using (3.1), Hn(s+u)=Hn(s)+Hn(u).

    (ii) If ϕn is an epimorphism:

    Rewriting (3.4) in the form

    ϕn(v)θHn(s,u,+)Z(Λ),

    i.e., ϕn(v)α=βZ(Λ), where α=θHn(s,u,+)Z(Λ), and βZ(Λ). Since ϕn is an epimorphism, we get Λα is an ideal contained in Z(Λ) and therefore Λα={0}. If K(Λ) is the two-sided annihilator of Λ, then, we have αK(Λ). But K(Λ) is an ideal contained in Z(Λ), so α=0 and using (3.1), Hn(s+u)=Hn(s)+Hn(u)

    Applying the previous theorem, when n=0, we obtain the following special case

    Corollary 3.1. Assume Λ is a ring. If Λ containing no non-zero ideals in the center, then every nilpotent CE-derivation D is additive.

    Also, when n=1, we get the case of ordinary CE-homoderivation as a special case.

    Corollary 3.2. Let Λ be any ring containing no non-zero ideals in the center. Then, every nilpotent CE-homoderivation H on Λ and every CE-homoderivation H on Λ related to an epimorphism ϕ1(t)=t+H1(t) is additive.

    Theorem 3.2. If the semiprime ring Λ has no non-zero ideals in its center, then each CE-n-homoderivation Hn on Λ related to an epimorphism ϕn is an n-homoderivation.

    Proof. Let s,u,tΛ be arbitrary elements. Then,

    Hn((su)t)nHn(su)Hn(t)suHn(t)Hn(su)tZ(Λ)andHn(s(ut))nHn(s)Hn(ut)sHn(ut)Hn(s)utZ(Λ). (3.9)

    Subtracting, we get

    Hn(su)ϕn(t)suHn(t)+ϕn(s)Hn(ut)+Hn(s)utZ(Λ). (3.10)

    Let

    Hn(su)=ϕn(s)Hn(u)+Hn(s)u+θHn(s,u,),θHn(s,u,)Z(Λ)andHn(ut)=Hn(u)ϕn(t)+uHn(t)+θHn(u,t,),θHn(u,t,)Z(Λ). (3.11)

    Using (3.11) in (3.10), we obtain

    {ϕn(s)Hn(u)+Hn(s)u+θHn(s,u,)}ϕn(t)suHn(t)+ϕn(s){Hn(u)ϕn(t)+uHn(t)+θHn(u,t,)}+Hn(s)utZ(Λ),

    which can simplify to

    θHn(s,u,)ϕn(t)+ϕn(s)θHn(u,t,)Z(Λ). (3.12)

    This gives

    [ϕn(s)θHn(u,t,),ϕn(t)]=[ϕn(s),ϕn(t)]θHn(u,t,)=0.

    Since ϕn is an epimorphism, we have

    [s,ϕn(t)]θHn(u,t,)=0,for alls,t,uΛ. (3.13)

    Replacing s by sr, rΛ, and using (3.13) and (3.11) we have

    [s,ϕn(t)]r{Hn(ut)nHn(u)Hn(t)uHn(t)Hn(u)t}=0,for allr,s,t,uΛ. (3.14)

    Thus,

    [s,ϕn(t)]Λ{Hn(ut)nHn(u)Hn(t)uHn(t)Hn(u)t}={0}. (3.15)

    Presume that the ring Λ has a collection of prime ideals {KλλΩ} such that Kλ={0}, and let K denote a typical Kλ. Let ¯Λ=Λ/K and ¯Z(Λ) the center of ¯Λ, and let ¯r=r+K be a typical element of ¯Λ. Fix u and t above, and let s vary. Then θHn(u,t,) is fixed but θHn(s,u,) depends on s. As seen from (3.15), either

    (i) [s,ϕn(t)]K for all sΛ,

    or

    (ii) θHn(u,t,)=Hn(ut)Hn(u)Hn(t)uHn(t)Hn(u)tK,

    hence ¯ϕn(t)¯Z(Λ) or ¯θHn(u,t,)=¯0. It follows from (3.12) that for each sΛ, ¯θHn(s,u,) ¯ϕn(t)+¯ϕn(s) ¯θHn(u,t,) ¯Z(Λ) so that if ¯ϕn(t)¯Z(Λ), ¯Λ¯θHn(u,t,)¯Z(Λ). On the other hand, if ¯θHn(u,t,)=¯0, Certainly, it is true that ¯Λ¯θHn(u,t,)¯Z(Λ). Thus [rθHn(u,t,),u]K for all r,uΛ; and since Kλ={0}. This provides the conclusion that ΛθHn(u,t,) is a central ideal of Λ, therefore ΛθHn(u,t,)={0}. Thus, letting K(Λ) be the two-sided annihilator of Λ, we have θHn(u,t,)K(Λ). However, K(Λ) is a central ideal, so θHn(u,t,)=0. Since Hn is additive by Theorem 3.1, then Hn is an n-homoderivation.

    Corollary 3.3. Every CE-homoderivation H on Λ related to an epimorphism ϕ1(t)=t+H1(t), for each tΛ, is also a homoderivation if the only central ideal in the semiprime ring is the zero ideal.

    Corollary 3.4. Every CE-derivation D on Λ is also a homoderivation if the only central ideal in the semiprime ring is the zero ideal.

    Theorem 3.2, Examples 2.1 and 2.2 together provide the following result.

    Theorem 3.3. A semiprime ring Λ admits a CE-n-homoderivation Hn on Λ related to an epimorphism ϕn which is not an n-homoderivation if and only if the only ideal in the center of Λ is the zero ideal.

    Theorem 3.4. If a semiprime ring Λ has no non-zero central ideals, then every nilpotent CE-n-homoderivation Hn on Λ must be an n-homoderivation.

    Proof. Theorem 3.1 states that Hn is additive. For any u,s and t in Λ. From (3.12), we have

    θHn(u,s,)(t+nHn(t))θHn(s,t,)(u+nHn(u))Z(Λ). (3.16)

    Therefore,

    θHn(u,s,)[t+nHn(t),u+nHn(u)]=0. (3.17)

    Replacing t by Hk1n(t) in (3.17), we have

    θHn(u,s,)[Hk1n(t),u+nHn(u)]=0. (3.18)

    Replacing t by Hk2n(t) in (3.17) and using (3.18), we have

    θHn(u,s,)[Hk2n(t),u+nHn(u)]=0. (3.19)

    By repeating the previous procedures, we obtain

    θHn(u,s,)[Hn(t),u+nHn(u)]=0. (3.20)

    From (3.17) and (3.20), we obtain

    θHn(u,s,)[t,u+nHn(u)]=0. (3.21)

    Substituting tx for t in (3.21), we obtain

    θHn(u,s,)t[x,u+nHn(u)]=0.

    Therefore,

    θHn(u,s,)Λ[x,nHn(u)+u]={0}.

    Let K={KλΩΛ,Kλbe a prime ideal in Λ} and Kλ={0}. Suppose that K represents a standard Kλ in K. For each uΛ, we have either θHn(u,s,.)K,for allsΛ or [x,nHn(u)+u]K,for allxΛ. First, if θHn(u,s,.)K,for allsΛ, then K+θHn(u,s,.)=K,for allsΛ. Thus, K+ΛθHn(u,s,.)=K,for allsΛ. So, (K+ΛθHn(u,s,.))(K+r)=(K+r)(K+ΛθHn(u,s,.)),for alls,rΛ. Therefore, K+[ΛθHn(u,s,.),r]=K,for alls,rΛ. Thus, [ΛθHn(u,s,.),r]Kλ={0},for alls,rΛ. That is ΛθHn(u,s,.)Z(Λ),for allsΛ. So, θHn(u,s,.)=0for allsΛ. In the other case, if [x,nHn(u)+u]K,for each xΛ, then [x,nHn(u)+u]+K=K,for each xΛ. Therefore,

    [x+K,(nHn(u)+u)+K]=K, for each xΛ. (3.22)

    From (3.16) and (3.22), we have

    K=[θHn(u,s,.)(nHn(t)+t)+KθHn(s,t,.)(nHn(u)+u)+K,x+K]=[θHn(u,s,.)(nHn(t)+t)+K,x+K] for each s,t,xΛ. (3.23)

    As above in Eq (3.17) we get K=[θHn(u,s,.)t+K,x+K]=[θHn(u,s,.)t,x]+K,for each s,t,xΛ. Thus, [θHn(u,s,.)t,x]K,for each s,t,xΛ. Thus, we achieve [θHn(u,s,.)t,x]Kλ={0},for each u,s,t,xΛ. Again, θHn(u,s,.)=0,for allsΛ. Moreover, we have θHn(u,s,.)=0,for allu,sΛ. From (3.11), we have

    Hn(us)=Hn(u)s+uHn(s)+nHn(u)Hn(s).

    Therefore, Hn is an n-homoderivation of Λ.

    Corollary 3.5. Any nilpotent CE-homoderivation is also a homoderivation if the only central ideal in the semiprime ring is the zero ideal.

    Corollary 3.6. Any nilpotent CE-derivation is also a derivation if the only central ideal in the semiprime ring is the zero ideal.

    Theorem 3.4, Examples 2.1 and 2.2 together provide the following result

    Theorem 3.5. A semiprime ring Λ admits a CE-n-homoderivation Hn on Λ which is not an n-homoderivation if and only if Λ contains a non-zero ideal that is a subset of its center.

    A map F:ΛΛ preserves the subset SΛ if F(S)S. Our purpose of this section is to study preservation of Z(Λ) by CE-n-homoderivations. It is necessary to show that not all CE-n-homoderivations preserve Z(Λ). Here is an example for a CE-n-homoderivation, with Hn(Z(Λ))Z(Λ).

    Example 4.1. Let Λ2 be a noncommutative ring satisfying Λ22Z(Λ2), for example a noncommutative ring with Λ32={0}. Let Λ1 be a zero ring with (Λ1,+)(Λ2,+). Let f:(Λ1,+)(Λ2,+) be an isomorphism. Let Λ=Λ1Λ2, and let Hn:ΛΛ given by Hn((x,y))=(0,f(x)), where xΛ1,yΛ2. It is clear that Z(Λ)=(Λ1,Z(Λ2)). Thus, Hn is a CE-n-homoderivation, but Hn(Z(Λ)) is generally not central unless f(x) is zero. Moreover, Λ1{0} is a two-sided ideal in Λ, and Λ1{0}Z(Λ), but Hn(Λ1{0})Z(Λ).

    A CE-n-homoderivation preserves the center under certain conditions, according to the following theorem.

    Theorem 4.1. Let Λ be a ring with center Z(Λ), and assume that zero is the only nilpotent element in Z(Λ). Then every CE-n-homoderivation Hn on Λ associated with an epimorphism ϕn, or every nilpotent CE-n-homoderivation Hn on Λ, preserves Z(Λ).

    Proof. (i) The first case, when Hn is related to an epimorphism ϕn.

    Let ξZ(Λ) and rΛ. Then

    Hn(ξr)nHn(ξ)Hn(r)Hn(ξ)rξHn(r)Z(Λ)

    and

    Hn(rξ)nHn(r)Hn(ξ)Hn(r)ξrHn(ξ)Z(Λ),

    and by subtracting, we obtain

    [ϕn(r),Hn(ξ)]=[r,Hn(ξ)]+[nHn(r),Hn(ξ)]Z(Λ)forallrΛ. (4.1)

    Since ϕn is an epimorphism on Λ, we get

    [r,Hn(ξ)]Z(Λ)forallrΛ. (4.2)

    Replacing r by rHn(ξ) in (4.2) gives [r,Hn(ξ)]Hn(ξ)Z(Λ), so

    [[r,Hn(ξ)]Hn(ξ),r]=0=[r,Hn(ξ)]2forallrΛ. (4.3)

    Since there is no nontrivial nilpotent elements in Z(Λ), (4.2) and (4.3) give [r,Hn(ξ)]=0 for all rΛ, i.e., Hn(ξ)Z(Λ).

    (ⅱ) Now, we are in a position to prove the second case when Hn is nilpotent.

    From (4.1), we have

    [nHn(r)+r,Hn(ξ)]Z(Λ),for allrΛ. (4.4)

    Putting Hk1n(r) instead of r in (4.4), we get

    [Hk1n(r),Hn(ξ)]Z(Λ),for allrΛ. (4.5)

    Once more, substituting Hk2n(r) for r in (4.4) and using (4.5), we achieve

    [Hk2n(r),Hn(ξ)]Z(Λ), for each rΛ. (4.6)

    Using the same procedure as before, we get

    [Hn(r),Hn(ξ)]Z(Λ), for each rΛ. (4.7)

    From (4.4) and (4.7) we have

    [r,Hn(ξ)]Z(Λ), for each rΛ. (4.8)

    In (4.8), replacing r with rHn(ξ) gives

    [rHn(ξ),Hn(ξ)]=[r,Hn(ξ)]Hn(ξ)Z(Λ), for each rΛ. (4.9)

    Thus, we get [[r,Hn(ξ)]Hn(ξ),r]=0,for allrΛ. Therefore,

    [r,Hn(ξ)]2=0, for each rΛ. (4.10)

    However, the nilpotent elements in the center Z(Λ) are zero, so we can deduce that [r,Hn(ξ)]=0,for allrΛ from (4.8) and (4.10). Hence, Hn(ξ)Z(Λ), i.e., Hn preserves the center.

    Naturally, the following consequence follows.

    Corollary 4.1. Let Λ be a ring with center Z(Λ) that has no non-zero nilpotent central elements. Then every CE-homoderivation H on Λ associated with an epimorphism ϕn, or every nilpotent CE-derivation D on Λ, preserves Z(Λ).

    CE-n-homoderivations that preserve Z(Λ) may also preserve subsets of Z(Λ), namely the set K(Λ)={ξZ(Λ)ξΛZ(Λ)}. It is readily seen that K(Λ) is the maximal central ideal, a central ideal that contains all other central ideals.

    Theorem 4.2. If Hn is a CE-n-homoderivations on a ring Λ which preserves Z(Λ), then Hn preserves K(Λ).

    Proof. Let ξK(Λ). Since K(Λ)Z(Λ), Hn(ξ)Z(Λ). For arbitrary sΛ,

    Hn(ξs)nHn(ξ)Hn(s)ξHn(s)Hn(ξ)sZ(Λ);

    and since Hn(ξs)Z(Λ), Hn(ξ)Hn(s)Z(Λ), and ξHn(s)Z, and Hn(ξ)sZ(Λ). Therefore Hn(ξ)K(Λ).

    Corollary 4.2. Every CE-homoderivation H or every CE-derivation D on a ring Λ that preserves Z(Λ), then Hn and D preserve K(Λ).

    In this section, our main objective is to illustrate the requirements that ensure a prime or semiprime ring is commutative when it admits a CE-n-homoderivation.

    Theorem 5.1. If Hn is not an nhomoderivation of a prime ring Λ, then Λ is commutative.

    Proof. If Λ includes no non-zero central ideals, according to Theorem 3.4, Hn is an nhomoderivation on Λ, which is a contradiction. As a consequence, Λ has a non-zero ideal that is contained in the center Z(Λ). Thus, Λ is commutative using [14, Lemma 1(b)].

    Theorem 5.2. Let Λ be a prime ring and Hn be a CE-n-homoderivation. If Hn(0)0, then Λ is commutative.

    Proof. Let Hn be a CE-n-homoderivation with Hn(0)0. Since Hn(0+0)Hn(0)Hn(0)Z(Λ), we have Hn(0)Z(Λ). Since Hn(0t)nHn(0)Hn(t)Hn(0)t0Hn(t)Z(Λ), we now get Hn(0)ϕn(t)Z(Λ) for all tΛ. But ϕn(t) is epimorphism of Λ, then we get Hn(0)tZ(Λ) for all tΛ. Therefore, [Hn(0)t,v]=0,for allt,vΛ. Since Hn(0)Z(Λ), we get Hn(0)[t,v]=0, for all v,tΛ. Replacing t by wt, we arrive at Hn(0)w[t,v]=0, for each v,t,wΛ. So, Hn(0)Λ[t,v]=0,for all v,tΛ. Using the primeness of Λ and Hn(0)0, [t,v]=0,for all v,tΛ, i.e., Λ is commutative.

    Theorem 5.3. Let Λ be a prime ring endowed with either a non-zero nilpotent CE-n-homoderivation Hn, or a non-zero CE-n-homoderivation Hn associated with an epimorphism ϕn. If Hn([u,s])=0 or Hn(us)=0, for each u,sΛ, then Λ is commutative.

    Proof. If Λ has a non-zero central ideal, then by [14, Lemma 1(b)] Λ is commutative. Now, assume that the only central ideal in Λ is the zero ideal. Due to Theorem 3.1, Hn is additive. First, assume that Hn([u,s])=0, for allu,sΛ. Substituting su for u, we get Hn([su,s])=0=Hn(s[u,s]),for each u,s in Λ. Thus, we get

    Hn(u)[s,u]Z(Λ),for allu,sΛ. (5.1)

    In (5.1), putting su instead of s, the result is Hn(u)[s,u]uZ(Λ),for allu,sΛ. Thus,

    [t,Hn(u)[s,u]u]=0,for allu,s,tΛ,

    which leads to

    Hn(u)[s,u][t,u]=0,for allu,s,tΛ. (5.2)

    Putting tw in place t in (5.2) and using (5.2), we get

    Hn(u)[s,u]t[w,u]=0,for allu,s,w,tΛ. (5.3)

    Using the primeness of Λ, for each uΛ either uZ(Λ) or Hn(u)[s,u]=0,for allsΛ. Assume that uΛ with Hn(u)[s,u]=0for allsΛ. Replacing s by st, we get Hn(u)s[t,u]=0,for allt,sΛ. Thus, for each uΛ either uZ(Λ) or Hn(u)=0. Consider that

    A={uΛ:uZ(Λ)},

    and

    B={uΛ:Hn(u)=0}.

    Then, (A,+) and (B,+) are additive subgroups of the group (Λ,+), and the union of A and B gives the whole ring Λ. So either A=Λ implies Λ is commutative or B=Λ implies Hn=0.

    Second, let Hn(us)=0, for all u,s in Λ. Putting su instead of u in Hn(us)=0, then Hn(sus)=Hn(s(us))=0for allu,sΛ. So,

    Hn(s)(us)Z(Λ),for allu,sΛ. (5.4)

    Substituting us for u in (5.4), we get

    Hn(s)(us)sZ(Λ),for allu,sΛ.

    By [15, Lemma 4] for each sΛ, either Hn(s)(us)=0for alluΛ or sZ(Λ). Assume that sΛ where

    Hn(s)(us)=0for alluΛ. (5.5)

    Putting tu instead of u in (5.5) and using (5.5), we get Hn(s)t[u,s]=0for alluΛ. By the primeness of Λ, either Hn(s)=0 or sZ(Λ). Therefore, for each sΛ, there are two cases: Either Hn(s)=0 or sZ(Λ). Thus, Hn=0 or Λ is commutative.

    Theorem 5.4. Let Λ be a semiprime ring and K a non-zero left ideal of Λ. If Λ admits a CE-n-homoderivation, which is non-zero on K and centralizing on K, then Λ contains a non-zero central ideal.

    Proof. By Theorem 3.3, Λ has a non-zero central ideal or Hn is an n-homoderivation; and if Hn is an n-homoderivation, our theorem reduces to Tammam et al (2022), Theorem 2, which was an extension to Bell and Martindale [16] (1987), Theorem 3.

    As a demonstration of our findings, we achieve the subsequent result:

    Corollary 5.1. A prime ring Λ with either a nilpotent CE-homoderivation H or a nilpotent CE-derivation D is commutative if any of the following conditions hold.

    (1) Hn is not a homoderivation.

    (2) Hn(0) is not zero.

    (3) Hn([u,t])=0 (or Hn(ut)=0) for each u,tΛ.

    It is essential that a semiprime ring Λ be commutative if it admits a derivation D such that [s,t]=[D(t),D(s)], for all s,tΛ. we conclude with a commutativity theorem with hypotheses using CE-n-homoderivations. (For further details, see [17], Theorem 3.3; [18], Corollary 1.3.)

    Theorem 5.5. Let Λ be a semiprime ring and Hn a CE-n-homoderivation on Λ such that [u,t]=[Hn(t),Hn(u)] for all u,tΛ. If Hn is centralizing CE-n-homoderivation on Λ related with an epimorphism ϕn or Hn is nilpotent, then Λ is commutative.

    Proof. (i) If Hn is centralizing, then by Lemma 1.1, Hn is commuting. Thus, we have

    [Hn(t),t]=0foralltΛ. (5.6)

    Now, our assumption assert that

    [u,t]=[Hn(t),Hn(u)]forallu,tΛ. (5.7)

    Replacing u by tu in (5.7) and using (5.6) and (5.7), we obtain

    Hn(t)[ϕn(u),Hn(t)]=0forallu,tΛ.

    Since ϕn is surjective, we obtain

    Hn(t)[u,Hn(t)]=0forallu,tΛ. (5.8)

    We now replace u by uw in (5.8), thereby obtaining

    Hn(t)u[w,Hn(t)],=0forallu,t,wΛ.

    i.e.,

    [Hn(t),w]Λ[Hn(t),w]={0};

    and Λ is semiprime, gives

    [w,Hn(t)]=0forallw,tΛ. (5.9)

    Hence Hn(Λ)Z(Λ) and therefore Λ is commutative by (5.7).

    (ii) The second case, if Hn is nilpotent:

    Replacing u by tu in (5.7) and using (5.7), we obtain

    [Hn(t),u+nHn(u)]Hn(t)+Hn(u)[Hn(t),t]=0forallu,tΛ. (5.10)

    In (5.10), replacing u by Hk1n(u), we obtain

    [Hn(t),Hk1n(u)]Hn(t)=0forallu,tΛ, (5.11)

    using (5.7), gives

    [u,Hk2n(t)]Hn(t)=0forallu,tΛ, (5.12)

    replacing u by uw, gives

    [u,Hk2n(t)]wHn(t)=0forallu,tΛ, (5.13)

    replacing w by wHk1n(u), gives

    [u,Hk2n(t)]wHk1n(u)Hn(t)=0forallu,tΛ. (5.14)

    Commuting (5.13) with Hk1n(u), we get

    [[u,Hk2n(t)]wHn(t),Hk1n(u)]=0, (5.15)

    which gives

    [u,Hk2n(t)]w[Hn(t),Hk1n(u)]+[[u,Hk2n(t)]w,Hk1n(u)]Hn(t)=0. (5.16)

    using (5.13) and (5.14) in (5.16), we get

    [u,Hk2n(t)]w[Hn(t),Hk1n(u)]=0. (5.17)

    using (5.7) in (5.17), we obtain

    [u,Hk2n(t)]w[u,Hk2n(t)]=0. (5.18)

    By semi-primness of Λ, we obtain

    [u,Hk2n(t)]=0. (5.19)

    Thus, Hk2n(t)Z(Λ). Now, in (5.7), replacing t by Hk3n(t), we get Hk3n(t)Z(Λ). We repeat this until we get [u,t]=0, which gives the commutativity of Λ.

    We conclude the article by presenting the following open question: Can the results derived in this manuscript be extended to a more general framework, such as non-associative structures, specifically alternative rings and algebras? For recent publications in this area, refer to [19,20,21].

    The commutativity of a ring Λ with a special class of mappings known as centrally extended n-homoderivations, where n is an integer, is investigated in this article. The ideas of derivations and homoderivations are expanded upon by these maps. We also looked into certain characteristics of the center of these rings.

    M. S. Tammam: Conceptualization, methodology, validation, formal analysis, investigation, data curation, writing-original draft preparation, writing-review and editing, supervision; M. Almulhem: Validation, formal analysis, writing-review and editing, supervision. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare no conflicts of interest.



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