Research article

A new approach to the study of fixed points based on soft rough covering graphs

  • Mathematical approaches to structure model problems have a significant role in expanding our knowledge in our routine life circumstances. To put them into practice, the right formulation, method, systematic representation, and formulation are needed. The purpose of introducing soft graphs is to discretize these fundamental mathematical ideas, which are inherently continuous, and to provide new tools for applying mathematical analysis technology to real-world applications including imperfect and inexact data or uncertainty. Soft rough covering models (briefly, SRC-Models), a novel theory that addresses uncertainty. In this present paper, we have introduced two new concepts Li-soft rough covering graphs (Li-SRCGs) and the concept of fixed point of such graphs. Furthermore, we looked into a some algebras that dealt with the fixed points of Li-SRCGs. Applications of the algebraic structures available in covering soft sets to soft graphs may reveal new facets of graph theory.

    Citation: Imran Shahzad Khan, Nasir Shah, Abdullah Shoaib, Poom Kumam, Kanokwan Sitthithakerngkiet. A new approach to the study of fixed points based on soft rough covering graphs[J]. AIMS Mathematics, 2023, 8(9): 20415-20436. doi: 10.3934/math.20231041

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  • Mathematical approaches to structure model problems have a significant role in expanding our knowledge in our routine life circumstances. To put them into practice, the right formulation, method, systematic representation, and formulation are needed. The purpose of introducing soft graphs is to discretize these fundamental mathematical ideas, which are inherently continuous, and to provide new tools for applying mathematical analysis technology to real-world applications including imperfect and inexact data or uncertainty. Soft rough covering models (briefly, SRC-Models), a novel theory that addresses uncertainty. In this present paper, we have introduced two new concepts Li-soft rough covering graphs (Li-SRCGs) and the concept of fixed point of such graphs. Furthermore, we looked into a some algebras that dealt with the fixed points of Li-SRCGs. Applications of the algebraic structures available in covering soft sets to soft graphs may reveal new facets of graph theory.



    The algebra of derivations and generalized derivations play a crucial role in the study of functional identities and their applications. There are many generalizations of derivations viz., generalized derivations, multiplicative generalized derivations, skew generalized derivations, bgeneralized derivations, etc. The notion of bgeneralized derivation was first introduced by Koșan and Lee [17]. The most important and systematic research on the bgeneralized derivations have been accomplished in [11,17,22,26] and references therein. In this manuscript, we characterize bgeneralized derivations which are strong commutative preserving (SCP) on R. Moreover, we also discuss and characterize bgeneralized derivations involving certain differential/functional identities on rings possessing involution.

    In the early nineties, after a memorable work on the structure theory of rings, a tremendous work has been established by Brešar considering centralizing mappings, commuting mappings, commutativity preserving (CP) mappings and strong commutativity preserving (SCP) mappings on some appropriate subset of rings. Since then it became a tempting research idea in the matrix theory/operator theory/ring theory for algebraists. Commutativity preserving (CP) maps were introduced and studied by Watkins [32] and further extended to SCP by Bell and Mason [6]. Inspired by the concept of SCP maps, Bell and Daif [5] demonstrated the commutativity of (semi-)prime rings possessing derivations and endomorphisms on right ideals (see also [27] and references therein). In [12], Deng and Ashraf studied strong commutativity preserving maps in more general context as follows: Let R be a semiprime ring. If R admits a mapping ψ and a derivation δ on R such that [ψ(r),δ(v)]=[r,v] for every r,vR, then R is commutative. In 1994, Brešar and Miers [7] characterized an additive map f:RR satisfying SCP on a semiprime ring R and showed that f is of the form f(r)=λr+μ(r) for every rR, where λC, λ2=1 and μ:RR is an additive map. Later, Lin and Liu [23] extended this result to Lie ideals of prime rings. Chasing to this, several techniques have been developed to investigate the behavior of strong commutativity preserving maps (SCP) using restrictions on polynomials, invoking derivations, generalized derivations, etcetera. An account of work has been done in the literature [3,10,12,13,15,20,21,23,24,25,27,30,31].

    On the other hand, the study of additive maps on rings possessing involution was initiated by Brešar et al. [7] and they characterized the additive centralizing mappings on the skew-symmetric elements of prime rings possessing involution. In the same vein, Lin and Liu [24] describe SCP additive maps on skew-symmetric elements of prime rings possessing involution. Later, Liau et al. [21] improved the above mentioned result for non-additive SCP mappings. Interestingly, in 2015 Ali et al. [1] studied the SCP maps in different way on rings possessing involution. They established the commutativity of prime ring of charateristic not two possessing second kind of involution satisfying [δ(r),δ(r)][r,r]=0 for every rR, where δ is a nonzero derivation of R. Recently, Khan and Dar [8] improved this result by studying the case of generalized derivations.

    Motivated by the above presented work, in this manuscript we have characterized bgeneralized derivations on prime rings possessing involution. Moreover, we also present some examples in support of our main theorems.

    Throughout the manuscript unless otherwise stated, R is a prime ring with center Z(R), Q is the maximal right ring of quotients, C=Z(Q) is the center of Q usually known as the extended centroid of R and is a field. "A ring R is said to be 2 torsion free if 2r=0 (where rR) implies r=0". The characteristic of R is represented by char(R). "A ring R is called a prime ring if rRv=(0) (where r,vR) implies r=0 or v=0 and is called a semiprime ring in case rRr=(0) implies r=0". "An additive map rr of R into itself is called an involution if (i) (rv)=vr and (ii) (r)=r hold for all r,vR. A ring equipped with an involution is known as a ring with involution or a ring. An element r in a ring with involution is said to be hermitian/symmetric if r=r and skew-hermitian/skew-symmetric if r=r". The sets of all hermitian and skew-hermitian elements of R will be denoted by W(R) and K(R), respectively. "If R is 2torsion free, then every rR can be uniquely represented in the form 2r=h+k where hW(R) and kK(R). Note that in this case r is normal, i.e., rr=rr, if and only if h and k commute. If all elements in R are normal, then R is called a normal ring". An example is the ring of quaternions. "The involution is said to be of the first kind if Z(R)W(R), otherwise it is said to be of the second kind. In the later case it is worthwhile to see that K(R)Z(R)(0)". We refer the reader to [4,14] for justification and amplification for the above mentioned notations and key definitions.

    For r,vR, the commutator of r and v is defined as [r,v]=rvvr. We say that a map f:RR preserves commutativity if [f(r),f(v)]=0 whenever [r,v]=0 for all r,vR. Following [7], "let S be a subset of R, a map f:RS is said to be strong commutativity preserving (SCP) on S if [f(r),f(v)]=[r,v] for all r,vS". Following [33], "an additive mapping T:RR is said to be a left (respectively right) centralizer (multiplier) of R if T(rv)=T(r)v (respectively T(rv)=rT(v)) for all r,vR. An additive mapping T is called a centralizer in case T is a left and a right centralizer of R". In ring theory it is more common to work with module homomorphisms. Ring theorists would write that T:RR is a homomorphism of a ring module R into itself. For a prime ring R all such homomorphisms are of the form T(r)=qr for all rR, where qQ (see Chapter 2 in [4]). "An additive mapping δ:RR is said to be a derivation on R if δ(rv)=δ(r)v+rδ(v) for all r,vR". It is well-known that every derivation of R can be uniquely extended to a derivation of Q. "A derivation δ is said to be Qinner if there exists αQ such that δ(r)=αrrα for all rR. Otherwise, it is called Qouter (δ is not inner)". "An additive map H:RR is called a generalized derivation of R if there exists a derivation δ of R such that H(rv)=H(r)v+rδ(v) for all r,vR". The derivation δ is uniquely determined by H and is called the associated derivation of H. The concept of generalized derivations covers the both the concepts of a derivation and a left centralizer. We would like to point out that in [19] Lee proved that "every generalized derivation can be uniquely extended to a generalized derivation of Q and thus all generalized derivations of R will be implicitly assumed to be defined on the whole Q".

    The recent concept of generalized derivations were introduced by Koșan and Lee [17], namely, bgeneralized derivations which was defined as follows: "An additive mapping H:RQ is called a (left) bgeneralized derivation of R associated with δ, an additive map from R to Q, if H(rv)=H(r)v+brδ(v) for all r,vR, where bQ". Also they proved that if "R is a prime ring and 0bQ, then the associated map δ is a derivation i.e., δ(rv)=δ(r)v+rδ(v) for all r,vR". It is easy to see that every generalized derivation is a 1generalized derivation. Also, the mapping rRαr+brcQ for a,b,cQ is a bgeneralized derivation of R, which is known as inner bgeneralized derivation of R. In spite of this, they also characterized bgeneralized derivation. That is, every bgeneralized derivation H on a semiprime ring R is of the form H(r)=αr+bδ(r) for all rR, where a,bQ. Following important facts are frequently used in the proof of our results:

    Fact 2.1 ([1,Lemma 2.1]). "Let R be a prime ring with involution such that char(R)2. If K(R)Z(R)(0) and R is normal, then R is commutative."

    Fact 2.2. "Let R be a ring with involution such that char(R)2. Then every rR can uniquely represented as 2r=w+s, where wW(R) and sK(R)."

    Fact 2.3 ([17,Theorem 2.3]). "Let R be a semiprime ring, bQ, and let H:RQ be a bgeneralized derivation associated with δ. Then δ is a derivation and there exists αQ such that H(r)=αr+bδ(r) for every rR."

    Fact 2.4 ([8,Lemma 2.2]). "Let R be a non-commutative prime ring with involution of the second kind such that char(R)2. If R admits a derivation δ:RR such that [δ(w),w]=0 for every wW(R), then δ(Z(R))=(0)."

    We need a well-known lemma due to Martindale [28], stated below in a form, convenient for our purpose.

    Lemma 2.1 ([28,Theorem 2(a)]). "Let R be a prime ring and ai,bi,cj,djQ. Suppose that mi=1airbi+nj=1cjrdj=0 for all rR. If a1,,am are Cindependent, then each bi is a Clinear combination of d1,,dn. If b1,,bm are Cindependent, then each ai is a Clinear combination of c1,,cn."

    We need Kharchenko's theorem for differential identities to prime rings [16]. The lemma below is a special case of [16,Lemma 2].

    Lemma 2.2 ([16,Lemma 2]). "Let R be a prime ring and ai,bi,cj,djQ and δ a Qouter derivation of R. Suppose that mi=1airbi+nj=1cjδ(r)dj=0 for all rR. Then mi=1airbi=0=nj=1cjrdj for all rR."

    We begin with the following Lemma which is needed for developing the proof of our theorems:

    Lemma 3.1. Let R be a non-commutative prime ring of characteristic different from two with involuation of the second kind. If for any αR, [αr,αr][r,r]=0 for every rR, then αC and α2=1.

    Proof. For any rR, we have

    [αr,αr][r,r]=0. (3.1)

    This can also be written as

    α2[r,r]+α[r,α]r+α[α,r]r[r,r]=0 (3.2)

    for every rR. Replace r by r+s in above equation, where sK(R)Z(R), we get

    α2[r,r]+α[r,α]rα[r,α]s+α[α,r]r+α[α,r]s[r,r]=0 (3.3)

    for every rR and sK(R)Z(R). In view of (3.2), we have

    α[α,r]s+α[α,r]s=α[α,r+r]s=0 (3.4)

    for every rR and sK(R)Z(R). Since the involution is of the second kind, so we have

    α[α,r+r]=0 (3.5)

    for every rR. For r=w+s, where wW(R) and sK(R), observe that

    α[α,w]=0 (3.6)

    for every wW(R). Substitute ss for w in above expression, we get

    α[α,s]=0 (3.7)

    for every sK(R), since the involution is of the second kind. Observe from Fact 2.2 that 2α[α,r]=α[α,2r]=α[α,w+s]=α[α,w]+α[α,s]=0. Thus we have αC and, by our hypothesis, (α21)[r,r]=0, for every rR. By the primeness of R, it follows that either α2=1 or [r,r]=0, for any rR. Thus we are led to the required conclusion by Fact 2.1.

    Theorem 3.1. Let R be a non-commutative prime ring of characteristic different from two. If H is a non-zero bgeneralized derivation on R associated with a derivation δ on R and ψ is a non-zero map on R satisfying [ψ(r),H(v)]=[r,v] for every r,vR. Then there exists 0λC and an additive map μ:RC such that H(r)=λr, ψ(r)=λ1r+μ(r), for any rR.

    Proof. Notice that if either δ=0 or b=0, the map H reduces to a centralizer, that is H(v)=αv, for any vR. Then the conclusion follows as a reduced case of [25,Theorem 1.1]. Hence, in the rest of the proof we assume both b0 and δ0. By Fact 2.3, there exists αH such that H(r)=αr+bδ(r) for every rR. By the hypothesis

    [ψ(r),αv+bδ(v)]=[r,v] (3.8)

    for every r,vR. Taking of vz for v in above expression gives

    (αv+bδ(v))[ψ(r),z]+[ψ(r),bvδ(z)]=v[r,z] (3.9)

    for every r,v,zR.

    Suppose firstly that δ is not an inner derivation of R. In view of (3.9) and Lemma 2.2, we observe that

    (αv+bv)[ψ(r),z]+[ψ(r),bvz]=v[r,z] (3.10)

    for every r,v,z,v,zR. In particular, for v=0 we have

    bv[ψ(r),z]=0 (3.11)

    for every r,z,vR. By the primeness of R and since b0, it follows that ψ(r)Z(R), for any rR. On the other hand, by ψ(r)Z(R) and (3.8) one has that [r,v]=0 for any r,vR, which is a contradiction since R is not commutative.

    Therefore, we have to consider the only case when there is qH such that δ(r)=[r,q], for every rR. Thus we rewrite (3.10) as follows

    ((αbq)v+bvq)[ψ(r),z]+[ψ(r),bvzqbvqz]=v[r,z] (3.12)

    that is

    {(αbq)vψ(r)+bvqψ(r)ψ(r)bvqvr}z+{bqvαv}zψ(r)+ψ(r)bvzqbvzqψ(r)+vzr=0 (3.13)

    for every r,v,zR.

    Suppose there exists vR such that {bv,v} are linearly C-independent. From relation (3.13) and Lemma 2.1, it follows that, for any rR, both r and qψ(r) are C-linear combinations of {1,q,ψ(r)}. In other words, there exist α1,α2,α3,β1,β2,β3C, depending by the choice of rR, such that

    r=α1+α2ψ(r)+α3q (3.14)

    and

    qψ(r)=β1+β2ψ(r)+β3q. (3.15)

    Notice that, for α2=0, relation (3.14) implies that q commutes with element rR. On the other hand, in case α20, by (3.14) we have that

    ψ(r)=α12(rα1α3q). (3.16)

    Then, by using (3.16) in (3.15), it follows that

    β1+β3q=α12(qβ2)(rα1α3q). (3.17)

    So, by commuting (3.17) with q, we get α12(qβ2)[r,q]=0, implying that [r,q]=0 in any case.

    Therefore q commutes with any element of R and this contradicts the fact that δ0. Therefore, for any vR, {v,bv} must be linearly C-dependent. In this case a standard argument shows that bC, which implies that H(r)=(αbq)r+r(bq), for any rR. Hence H is a generalized derivation of R and once again the result follows from [25,Theorem 1.1].

    The following theorem is a generalization of [8,Theorem 2.3].

    Theorem 3.2. Let R be a non-commutative prime ring with involution of the second kind of characteristic different from two. If H is a bgeneralized derivation on R associated with a derivation δ on R such that [H(r),H(r)]=[r,r] for every rR, then there exists λC such that λ2=1 and H(r)=λr for every rR.

    Proof. By the given hypothesis, we have

    [H(r),H(r)][r,r]=0 for all rR. (3.18)

    Taking r as r+v in (3.18) to get

    [H(r),H(v)]+[H(v),H(r)][r,v][v,r]=0 (3.19)

    for every r,vR. Substitute vs for v, where sK(R)Z(R), in above relation, we obtain

    0=[H(r),H(v)]s[H(r),bv]δ(s)+[H(v),H(r)]s+[bv,H(r)]δ(s)+[r,v]s[v,r]s (3.20)

    for every r,vR and sK(R)Z(R). Multiply (3.19) with s and combine with (3.20) to get

    2[H(v),H(r)]s2[v,r]s[H(r),bv]δ(s)+[bv,H(r)]δ(s)=0 (3.21)

    for every r,vR and sK(R)Z(R). Again substitute v as vs in (3.21), we get

    0=2[H(v),H(r)]s2+2[bv,H(r)]δ(s)s2[v,r]s2+[H(r),bv]δ(s)s+[bv,H(r)]δ(s)s (3.22)

    for every r,vR and sK(R)Z(R). In view of (3.21), we have

    2[bv,H(r)]δ(s)s+2[H(r),bv]δ(s)s=0 (3.23)

    for every r,vR and sK(R)Z(R). Since char(R)2 and the involution is of the second kind, so [bv,H(r)]+[H(r),bv]=[H(r),bv][H(r),bv]=0 for every r,vR or δ(s)s=0 for every sK(R)Z(R). Observe that s=0 is also implies δ(s)=0 for every sK(R)Z(R). Assume that δ(s)0, therefore we have

    [H(r),bv][H(r),bv]=0 (3.24)

    for every r,vR. Taking r=v=w+s in above expression, we obtain

    [H(s),bw][H(w),bs]=0 (3.25)

    for every wW(R) and sK(R). Replace s by s in (3.25), we get

    [H(s),bw][H(w),b]s=0 (3.26)

    for every wW(R) and sK(R)Z(R). Substitute ss for w in last expression, we get

    [H(s),bs]s[H(s),b]s2+b[b,s]δ(s)s=0 (3.27)

    for every sK(R) and sK(R)Z(R). One can see from (3.24) that [H(s),bs]=0 and [H(s),b]s=0 for every sK(R) and sK(R)Z(R). This reduces (3.27) into

    b[b,s]δ(s)s=0 (3.28)

    for every sK(R) and sK(R)Z(R). This implies either b[b,s]=0 for every sK(R) or δ(s)=0 for every sK(R)Z(R). Suppose b[b,s]=0 for every sK(R). Take s=w0s and use the fact that the involution is of the second kind, we get b[b,w0]=0 for every w0W(R). An application of Fact 2.2 yields bC. One can see from (3.27) that b[H(s),s]=0 for every sK(R) and sK(R)Z(R). If b=0 and in light of Fact 2.3, H has the following form: H(r)=αr, for some fixed element αH. Thus, by Lemma 3.1 and since R is not commutative, we get the required conclusion αC and α2=1.

    So we assume b0 and [H(s),s]=0 for every sK(R) and sK(R)Z(R). Again taking s as w0s and making use of Fact 2.2 in last relation gives H(s)Z(R) for every sK(R)Z(R). Next, take r=w and v=s in (3.19), we get

    [H(s),H(w)]+[w,s]=0 (3.29)

    for every sK(R) and wW(R). Substitute w0s for s in above relation, we get

    [H(w0s),H(w)]+[w,w0]s=0 (3.30)

    for every sK(R)Z(R) and w,w0W(R). It follows from the hypothesis that

    [H(w0),H(w)]s+[bw0,H(w)]δ(s)+[w,w0]s=0 (3.31)

    for every sK(R)Z(R) and w,w0W(R). For w0=w, we have

    b[H(w),w]δ(s)=0 (3.32)

    for every sK(R)Z(R) and wW(R). Since δ(s)0 and b0, so [H(w),w]=0 for every wW(R). Since bC, so we observe from (3.25) that

    [H(s),w][H(w),s]=0 (3.33)

    for every wW(R) and sK(R). Repalcement of s by sh in above expression and making use of H(s)Z(R) for every sK(R)Z(R) and b0 yields [δ(w),w]=0 for every wW(R). In light of Fact 2.4, we have δ(s)=0 for every sK(R)Z(R). Finally, we suppose δ(s)=0 and substitute v by vs in (3.19), we obtain

    [H(r),H(v)]s+[H(v),H(r)]s+[r,v]s[v,r]s=0 (3.34)

    for every r,vR and sK(R)Z(R). Combination of (3.19) and (3.34) gives

    ([H(v),H(r)][v,r])s=0 (3.35)

    for every r,vR and sK(R)Z(R). This implies that

    [H(v),H(r)][v,r]=0 (3.36)

    for every r,vR. In particular

    [H(r),H(v)][r,v]=0 (3.37)

    for every r,vR. As a special case of Theorem 3.1, H is of the form H(r)=λr, where λC and λ2=1.

    The following theorem is a generalization of [8,Theorem 2.4].

    Theorem 3.3. Let R be a non-commutative prime ring with involution of the second kind of characteristic different from two. If H is a bgeneralized derivation on R associated with a derivation δ on R such that [H(r),δ(r)]=[r,r] for every rR, then there exists λC such that H(r)=λr for every rR.

    Proof. By the hypothesis, we have

    [H(r),δ(r)][r,r]=0 (3.38)

    for every rR. The derivation δ satisfies δ(R)Z(R), otherwise the hypothesis [H(r),δ(r)]=[r,r] for every rR, would reduce to [r,r]=0 for all rR, and, therefore R would be commutative, by Fact 2.1. A linearization of (3.38) yields

    [H(r),δ(v)]+[H(v),δ(r)][r,v][v,r]=0 (3.39)

    for every r,vR. Replace r by zZ(R) in (3.39), we obtain [H(z),δ(v)]=0 for every vR and zZ(R). Observe from [18,Theorem 2] that H(z)Z(R) for every zZ(R). Now take r as rs in (3.38) and suppose δ(s)0, we get

    0=[H(r),δ(r)]s2[H(r),r]δ(s)sb[r,δ(r)]δ(s)s[b,δ(r)]rδ(s)sb[r,r]δ(s)2[b,r]rδ(s)2+[r,r]s2 (3.40)

    for every rR and sK(R)Z(R). In view of (3.38), we have

    0=[H(r),r]δ(s)sb[r,δ(r)]δ(s)s[b,δ(r)]rδ(s)sb[r,r]δ(s)2[b,r]rδ(s)2 (3.41)

    for every rR and sK(R)Z(R). Replace r by r+s in last expression and use the fact that H(z)Z(R) for every zZ(R), we get

    0=[H(r),r]δ(s)sb[r,δ(r)]δ(s)s[b,δ(r)]rδ(s)sb[r,r]δ(s)2[b,r]rδ(s)2[b,δ(r)]δ(s)s2[b,r]δ(s)2s (3.42)

    for every rR and sK(R)Z(R). Observe from (3.41)

    [b,δ(r)]δ(s)s2+[b,r]δ(s)2s=0 (3.43)

    for every rR and sK(R)Z(R). Replace r by rs and use (3.43), we get [b,r]δ(s)2s=0 This forces that [b,r]=0 for every rR since δ(s)0. One can easily obtain from last relation that bC. On the other hand

    0=H(s)[r,δ(r)]sb[δ(r),δ(r)]s2[b,δ(r)]δ(r)s2H(s)[r,r]δ(s)b[δ(r),r]δ(s)s[b,r]δ(r)δ(s)s+[r,r]s2 (3.44)

    for every rR and sK(R)Z(R). Since bC, so we have

    0=H(s)[r,δ(r)]sb[δ(r),δ(r)]s2H(s)[r,r]δ(s)b[δ(r),r]δ(s)s+[r,r]s2 (3.45)

    for every rR and sK(R)Z(R). Now substitute w and s for r and v in (3.39), respectively. This yields

    [H(s),δ(w)][H(w),δ(s)]+2[s,w]=0 (3.46)

    for every wW(R) and sK(R). Take s as sw in last relation and use bC, we see that

    H(s)[w,δ(w)][H(w),w]δ(s)=0 (3.47)

    for every wW(R) and sK(R)Z(R). On the other hand

    b[h,δ(w)]δ(s)[H(w),w]δ(s)=0 (3.48)

    for every wW(R) and sK(R)Z(R). In view of (3.47) and (3.48), we get

    (H(s)+bδ(s))[w,δ(w)]=0 (3.49)

    for every wW(R) and sK(R)Z(R). Since H(s)Z(R), δ(s)Z(R) and bZ(R), so either H(s)+bδ(s)=0 for every sK(R)Z(R) or [w,δ(w)]=0 for every wW(R). If [w,δ(w)]=0 for every wW(R), then δ(s)=0 for every sK(R)Z(R) from Fact 2.4. Therefore consider H(s)=bδ(s) for every sK(R)Z(R) and use it in (3.45), we obtain

    0=b[r,δ(r)]δ(s)sb[δ(r),δ(r)]s2+b[r,r]δ(s)2b[δ(r),r]δ(s)s+[r,r]s2 (3.50)

    for every rR and sK(R)Z(R). For r=w, we have

    2b[h,δ(h)]δ(s)s=0 (3.51)

    for every wW(R) and sK(R)Z(R). Thus b=0 or [w,δ(w)]=0 for every wW(R) or δ(s)=0 for every sK(R)Z(R). If b=0 and since s is not a zero-divisor, the relation (3.50) reduces to [r,r]=0, for every rR. Thus the commutativity of R follows from Fact 2.1, a contradiction. The rest of two relations yields δ(s)=0 for every sK(R)Z(R). Finally consider δ(s)=0 and replace r by rs in (3.38) and use the facts that the involution is of the second kind, we see that

    [H(r),δ(v)][H(v),δ(r)][r,v]+[v,r]=0 (3.52)

    for every r,vR. Combination of (3.39) and (3.52) yields

    [H(r),δ(v)][r,v]=0 (3.53)

    for every r,vR. In particular,

    [H(r),δ(v)][r,v]=0 (3.54)

    for every r,vR. In view of Theorem 3.1, there exist 0λC and an additive map μ:RC such that H(r)=λr and δ(r)=λ1r+μ(r) for every rR, where δ=δ. Commute the latter case with r, we get [δ(r),r]=0 for every rR. Since δ=δ0, so R is commutative from [29,Lemma 3], this leads to again a contradiction. This completes the proof of the theorem.

    The following example shows that the condition of the second kind involution is essential in Theorems 3.2 and 3.3. This example collected from [2,Example 1].

    Example 4.1. Let

    R={(β1β2β3β4)|β1,β2,β3,β4Z},

    which is of course a prime ring with ususal addition and multiplication of matrices, where Z is the set of integers. Define mappings H,δ,:RR such that

    H(β1β2β3β4)=(0β2β30),
    δ(β1β2β3β4)=(0β2β30),

    and a fixed element

    b=(1001),
    (β1β2β3β4)=(β4β2β3β1).

    Obviously,

    Z(R)={(β100β1)|β1Z}.

    Then r=r for every rZ(R), and hence Z(R)W(R), which shows that the involution is of the first kind. Moreover, H, δ are nonzero bgeneralized derivation and associated derivation with fixed element b defined as above, such that the hypotheses in Theorems 3.2 and 3.3 are satisfied but H is not in the form H(r)=λr for every rR. Thus, the hypothesis of the second kind involution is crucial in our results.

    We conclude the manuscript with the following example which reveals that Theorems 3.2 and 3.3 cannot be extended to semiprime rings.

    Example 4.2. Let (R,) be a ring with involution as defined above, which admits a bgeneralized derivation H, where δ is an associated nonzero derivation same as above and R1=C with the usual conjugation involution . Next, let S=R×R1 and define a bgeneralized derivation G on S by G(r,v)=(H(r),0) associated with a derivation D defined by D(r,v)=(δ(r),0). Obviously, (S,τ) is a semiprime ring with involution of the second kind such that τ(r,v)=(r,v). Then the bgeneralized derivation G satisfies the requirements of Theorems 3.2 and 3.3, but G is not in the form G(r)=λr for every rR, and R is not commutative. Hence, the primeness hypotheses in our results is not superfluous.

    We recall that "a generalized skew derivation is an additive mapping G:RR satisfying the rule G(rv)=G(r)v+ζ(r)(v) for every r,vR, where is an associated skew derivation of R and ζ is an automorphism of R". Following [9], De Filippis proposed the new concept for further research and he defined the following: "Let R be an associative algebra, bQ, be a linear mapping from R to itself, and ζ be an automorphism of R. A linear mapping G:RR is called an Xgeneralized skew derivation of R, with associated term (b,ζ,) if G(rv)=G(r)v+bζ(r)(v) for every r,vR". It is clear from both definitions, the notions of Xgeneralized skew derivation, generalize both generalized skew derivations and skew derivations. Hence, every Xgeneralized skew derivation is a generalized skew derivation as well as a skew derivation, but the converse statement is not true in general.

    Actuated by the concept specified by De Filippis [9] and having regard to our main theorems, the following are natural problems.

    Question 5.1. Let R be a (semi)-prime ring and L be a Lie ideal of R. Next, let F and G be two Xgeneralized skew derivation with an associated skew derivation of R such that

    [F(r),G(v)]=[r,v],for everyr,vL.

    Then what we can say about the behaviour of F and G or about the structure of R?

    Question 5.2. Let R be a prime ring possessing second kind involution with suitable torsion restrictions and L be a Lie ideal of R. Next, let F and G be two Xgeneralized skew derivation with an associated skew derivation of R such that

    [F(r),G(r)]=[r,r],for everyrL.

    Then what we can say about the behaviour of F and G or about the structure of R?

    The characterization of strong commutative preserving (SCP) bgeneralized derivations has been discussed in non-commutative prime rings. In addition, the behavior of bgeneralized derivations with differential/functional identities on prime rings with involution was investigated. Besides, we present some problems for Xgeneralized skew derivations on rings with involution.

    We are very grateful to the referee for his/her appropriate and constructive suggestions which improved the quality of the paper. This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (G: 156-662-1439). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

    The authors declare that they have no competing interests.



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