Let R be a *-ring and n≥ 1 be an integer. The objective of this paper is to introduce the notion of n-skew centralizing maps on *-rings, and investigate the impact of these maps. In particular, we describe the structure of prime rings with involution '*' such that *[x,d(x)]n∈Z(R) for all x∈ R (for n=1, 2), where d:R→R is a nonzero derivation of R. Among other related results, we also provide two examples to prove that the assumed restrictions on our main results are not superfluous.
This research is motivated by the recent work's of Ali-Dar [1], Qi-Zhang [5] and Hou-Wang [3]. However, our approach is different from that of the authors of [5] and [3]. A ring R with an involution '*' is called a *-ring or ring with involution '*'. Throughout, we let R be a ring with involution '*' and Z(R), the center of the ring R. Moreover, the sets of all hermitian and skew-hermitian elements of R will be denoted by H(R) and S(R), respectively. The involution is called the first kind if Z(R)⊆H(R), otherwise S(R)∩Z(R)≠(0)(see [2] for details). A ring R is said to be 2-torsion free if 2x=0 (where x ∈ R) implies x=0. A ring R is called prime if aRb=(0) (where a,b ∈ R) implies a=0 or b=0. A derivation on R is an additive mapping d:R→R such that d(xy)=d(x)y+xd(y) for all x,y ∈ R.
For any x,y ∈ R, the symbol [x,y] will denote the Lie product xy-yx and the symbol *[x,y] will denote the skew Lie product xy-yx*, where '*' is an involution on R. In a recent paper, Hou and Wang [3] extended the concept of skew Lie product as follows: for an integer n ≥ 1, the n-skew Lie product of any two elements x and y is defined by *[x,y]n=*[x,*[x,y]n−1], where *[x,y]0=y, *[x,y]=xy−yx* and *[x,y]2=x2y−2xyx*+y(x*)2. Obviously, for n=1, the skew Lie product and n-skew Lie product coincides. Note that, for n=2, we call it 2-skew Lie product. In [3], Hou and Wang studied the strong 2-skew commutativity preserving maps in prime rings with involution. In fact, they described the form of strong 2-skew commutativity preserving maps on a unital prime ring with involution that contains a non-trivial symmetric idempotent. In [5], Qi and Zhang studied the properties of n-skew Lie product on prime rings with involution and as an application, they characterized n-skew commuting additive maps, i.e.; an additive mapping f on R into itself such that *[x,f(x)]n=0 for all x∈ R. In definition of n-skew commuting mapping (defined in [5]), if we consider that f is any map (not necessarily additive) then it is more reasonable to call f a n-skew commuting. To give its precise definition, we make a slight modification in Qi and Zhang's definition for n-skew commuting mapping. For an integer n ≥ 1, a map f of a *-ring R into itself is called n-skew commuting mapping on R if *[x,f(x)]n=0 for every x∈ R. For an integer n ≥ 1, a map f of a *-ring R into itself is called n-skew centralizing mapping on R if *[x,f(x)]n∈Z(R) for every x ∈ R. In particular, for n = 1, 2, we call them 1-skew commuting (resp. 1-skew centralizing) and 2-skew commuting (resp. 2-skew centralizing) mapping.
The objective of this paper is to introduce the notion of n-skew centralizing mappings on *-rings. Further, we investigate the impact of these mappings and describe the nature of prime *-rings which satisfy certain *-differential identities. In particular, for an integer n ≥ 1 we prove that if a 2-torsion free prime ring R with involution '*' of the second kind which admits a nonzero derivation d such that *[x,d(x)]n∈Z(R) for all x∈R(n=1,2), then R is commutative. Moreover, some more related results are obtained. Further more, examples prove that, the assumed curtailment can not be relaxed as given.
In order to study the effect of n-skew centralizing mappings, we need the following two lemmas for developing the proofs of our main results. We begin our discussion with the following lemmas:
Lemma 2.1. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If x2x*∈Z(R) for all x∈ R, then R is commutative.
Proof. Linearization of x2x*∈Z(R) gives that x2y*+xyx*+xyy*+yxx*+yxy*+y2x*∈Z(R) for all x,y∈R. Taking x=-x in the last expression and combine it with the above relation, we get
x2y*+xyx*+yxx*∈Z(R)for allx,y∈R.
Substituting ky for y, where k∈S(R)∩Z(R), we obtain (−x2y*+xyx*+yxx*)k∈Z(R) for all x,y∈R. Invoking the primeness of R and using the fact that S(R)∩Z(R)≠(0), we get
−x2y*+xyx*+yxx*∈Z(R)for allx,y∈R.
Combining (2.1) and (2.2), we conclude that x2y*∈Z(R). Replacing y by y*, we get x2y∈Z(R) for all x,y∈R. This can be further written as [x2y,w]=0 for all x,y,w∈ R. Replacing y by ry, we get x2r[y,w]=0 for all x,y,w,r∈R. Hence by the primeness of the ring R, we are force to conclude that R is commutative.
Lemma 2.2. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If *[x,x*]2∈Z(R) for all x ∈ R, then R is commutative.
Proof. By the hypothesis, we have
x2x*−2x(x*)2+(x*)3∈Z(R)for allx∈R.
Replacing x by kx in (2.3) where k∈S(R)∩Z(R), we get
−x2x*k3−2x(x*)2k3−(x*)3k3∈Z(R)for allx∈R.
By (2.3) and (2.4), we conclude that −4x(x*)2k3∈Z(R) for all x∈ R. Since R is 2-torsion free prime ring and S(R)∩Z(R)≠(0), we obtain x(x*)2∈Z(R) for all x∈ R. On linearizing we get
Substitute ky for y, where k∈S(R)∩Z(R) in (2.6), we get 2y(x*)2k∈Z(R) for all x,y∈R. This implies that yx2∈Z(R) for all x,y∈R. Henceforth, using the same arguments as we have used in Lemma 2.1, we conclude that R is commutative. This proves the lemma.
Theorem 2.3. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If R admits a nonzero derivation d such that *[x,d(x)]n∈Z(R) for all x∈R(n=1,2), then R is commutative.
proof.Case(i) First we discuss the case, "when n=1" and "i.e.",
Using the primeness of R, we obtain either d(h)=0 or
x[y,r]+[x,r]y−y[x*,r]−[y,r]x*=0for allx,y,r∈R.
First we consider the situation
x[y,r]+[x,r]y−y[x*,r]−[y,r]x*=0for allx,y,r∈R.
Substituting kx for x in (2.8), where k∈S(R)∩Z(R) and combining it with (2.8), we get
2(x[y,r]+[x,r]y)k=0for allx,y,r∈R.
Since R is 2-torsion free prime ring, we deduce that
x[y,r]+[x,r]y=0for allx,y,r∈R.
Replacing x by z, where z ∈ Z(R), we get [y,r]z=0 for all y,r ∈ R. Henceforth, we conclude that R is commutative. Now consider the case d(h)=0 for all h∈H(R)∩Z(R). This implies that d(k)=0 for all k∈S(R)∩Z(R). Replacing y by ky in (2.7), where k∈S(R)∩Z(R) with d(k)=0 and adding with (2.7), we get
for all x,y,r ∈ R. Since R is 2-torsion free ring and S(R)∩Z(R)≠(0), the above relation implies that
x[d(y),r]+[x,r]d(y)−d(y)[x*,r]−[d(y),r]x*+y[d(x),r]+[y,r]d(x)=0for all
x,y∈R. Taking y=h, where h∈H(R)∩Z(R) and using the fact that d(h)=0, we get [d(x),r]h=0 for all x,r ∈ R and h∈H(R)∩Z(R). This yields that [d(x),r]=0 for all x,r ∈ R. Hence in view of Posner's [4] first theorem, R is commutative.
Case (ii) Now, we prove the result for n=2 i.e.,
*[x,d(x)]2∈Z(R)for allx∈R.
On expansion we acquire
x2d(x)−2xd(x)x*+d(x)(x*)2∈Z(R)for allx∈R.
Replacing x by xh in (2.9), where h∈H(R)∩Z(R), we obtain
(x3−2x2x*+x(x*)2)d(h)h2∈Z(R)for allx∈R.
Then by the primeness of R we are force to conclude that either d(h)h2=0 or x3−2x2x*+x(x*)2∈Z(R) for all x∈ R. First we consider the case
x3−2x2x*+x(x*)2∈Z(R)for allx∈R.
Substituting kx for x in (2.10), where k∈S(R)∩Z(R), we get (x3+2x2x*+x(x*)2)k3∈Z(R)for allx∈R. This further implies that
x3+2x2x*+x(x*)2∈Z(R)for allx∈R.
Subtracting (2.10) from (2.11) and using 2-torsion freeness of R, we obtain x2x*∈Z(R)for allx∈R. Therefore, by Lemma 2.1, R is commutative. Now consider the second case d(h)h2=0 for all h∈H(R)∩Z(R). This implies that d(h)=0 for all h∈H(R)∩Z(R). Therefore, d(k)=0 for all k∈S(R)∩Z(R). Replacing x by kx in (2.9), where k∈S(R)∩Z(R) and using the fact that d(k)=0, we obtain
x2d(x)k3+2xd(x)x*k3+d(x)(x*)2k3∈Z(R)for allx∈R.
Application of (2.9) yields 4xd(x)x*k3∈Z(R) for all x ∈ R. Since R is 2-torsion free ring and S(R)∩Z(R)≠(0), we get
xd(x)x*∈Z(R)for allx∈R.
Putting x+h in place of x, where h∈H(R)∩Z(R), we arrive at
xd(x)h+d(x)x*h+d(x)h2∈Z(R)for allx∈R.
Taking x=-x in (2.14) and then combining it with the obtained relation, we get 2d(x)h2∈Z(R). This implies that d(x)∈Z(R) for all x∈ R, since the involution '*' is of the second kind. Hence, by Posner's [4] first theorem, R is commutative.
Theorem 2.4. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If R admits a nonzero derivation d such that d(*[x,x*]n)∈Z(R) for all x ∈ R(n=1,2), then R is commutative.
proof.Case (i) Firstly we are focus to discuss the case when n=1 i.e.,
for all x,y,r ∈ R. We first consider the relation (2.16). Replacing y by ky, where k∈S(R)∩Z(R) in (2.16), we get 2(y[x*,r]+[y,r]x*)k=0 for all x,y,r∈R. Since R is 2-torsion free ring and S(R)∩Z(R)≠(0), we obtain y[x*,r]+[y,r]x*=0 for all x,y,r∈R. Taking x=k, where k∈S(R)∩Z(R), we get -[y,r]k=0 for all y,r∈ R. Thus -[y,r]=0 for all y,r∈ R. That is, R is commutative. Now consider d(h)=0 for all h∈H(R)∩Z(R). This implies that d(k)=0 for all k∈S(R)∩Z(R). Replacing y by ky, where k∈S(R)∩Z(R) in (2.15) and making use of (2.15), we get
Taking x=h where h∈H(R)∩Z(R) and using d(h)=0, we arrive at h[d(y),r]=0 for all y,r ∈ R. Then by the primeness of R and the fact that S(R)∩Z(R)≠(0), we obtain [d(y),r]=0 for all y,r∈ R. Hence by Posner's [4] first theorem, R is commutative.
for all x,y ∈ R. Replacing y by ky, where k∈S(R)∩Z(R) in (2.20), we arrive at
2(xyx*+yxx*−2y(x*)2)∈Z(R)for allx,y∈R.
This implies that
xyx*+yxx*−2y(x*)2∈Z(R)for allx,y∈R.
Substituting kx for x in the last relation, we conclude that y(x*)2∈Z(R) for all x,y∈ R. Now proceed as we have already done in Lemma 2.1, we conclude that R is commutative. "Considering the second case in which we have d(h)=0 for all h∈H(R)∩Z(R)." This implies that d(k)=0 for all k∈S(R)∩Z(R). Now replacing x by kx in (2.19) and using "d(k)=0, we get"
Combining this with (2.19) and using the fact that R is 2-torsion free ring, we arrive at
d(xyx*)+d(yxx*)−2d(y(x*)2)∈Z(R)for allx,y∈R.
Replacing x by kx, where k∈S(R)∩Z(R) and combining it with previous expression, we obtain 2d(y(x*)2)∈Z(R) for all x,y ∈ R. Replacing x by h, where h∈H(R)∩Z(R) we come to d(y)h2∈Z(R) for all y ∈ R. This implies that d(y)∈Z(R) for all y ∈ R. Hence, R is commutative by Posner's [4] first theorem.
Theorem 2.5. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If R admits a derivation d such that *[x,d(x)]2+*[x,x*]2∈Z(R) for all x ∈ R, then R is commutative.
Proof. By the hypothesis we assume that
*[x,d(x)]2+*[x,x*]2∈Z(R)for allx∈R.
If we take d=0. Then, application of Lemma 2.2, yields the required result. Now consider the case d ≠ 0 and on expansion of (2.21), we get
Replacing x by xh in (2.21), where h∈H(R)∩Z(R), we obtain (x3−x2x*+x(x*)2)d(h)h2∈Z(R) for all x ∈ R. Now by the primeness of R we get either x3−2x2x*+x(x*)2∈Z(R) for all x ∈ R or d(h)h2=0. Now, we suppose that
x3−2x2x*+x(x*)2∈Z(R)for allx∈R.
This is same as the relation (2.10) in Theorem 2.3 and hence we conclude that R is commutative. Now we consider the case d(h)h2=0 for all h∈H(R)∩Z(R). Since R is prime ring, so we get d(h)=0. This also implies that d(k)=0 for all k∈S(R)∩Z(R). Replacing x by xk in (2.22) and combining with (2.22) we arrive at (2xd(x)x*−x2x*−(x*)3)k3∈Z(R) for all x∈ R. Since S(R)∩Z(R)≠(0), so by the primeness of R, we get
∈Z(R) for all x,y ∈ R. Taking x=h, where h∈H(R)∩Z(R), we get
(2d(y)−4y*−2y)h2∈Z(R)for ally∈R.
The primeness of R yields that d(y)−2y*−y∈Z(R) for all y ∈ R. Replacing y by ky and on solving, we get y ∈ Z(R) for all y ∈ R. Hence, this conclude that R is commutative.
Theorem 2.6. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If R admit two distinct derivations d1 and d2 such that *[x,d1(x)]2−*[x,d2(x)]2∈Z(R) for all x ∈ R, then R is commutative.
Proof. We assume that
*[x,d1(x)]2−*[x,d2(x)]2∈Z(R)for allx∈R.
If either d1 or d2 is zero, then we get the required result by Theorem 2.3 above. Now consider both d1, d2 are non-zero. Expansion of (2.26) yields that
for all x ∈ R. Replacing x by xh in (2.27), where h∈H(R)∩Z(R) and on simplifying with the help of (2.27), we get
(x3−2x2x*+x(x*)2)(d1(h)−d2(h))h2∈Z(R)for allx∈R.
This implies either x3−2x2x*+x(x*)2∈Z(R) for all x ∈ R or (d1(h)−d2(h))h2=0. If x3−2x2x*+x(x*)2∈Z(R) for all x∈ R, then by using the same steps as we have used after (2.10), we arrive at x2x*∈Z(R) for all x ∈ R. Thus R is commutative, by Lemma 2.1. On the other hand, if (d1(h)−d2(h))h2=0 for all h∈H(R)∩Z(R). Then we are force to conclude that d1(h)=d2(h) and hence d1(k)=d2(k) for all k∈S(R)∩Z(R). Replacing x by kx in (2.27), and combining with (2.27) by using the fact that d1(k)=d2(k), we get
4(xd1(x)x*−xd2(x)x*)k3∈Z(R)for allx∈R.
Since R is 2-torsion free and S(R)∩Z(R)≠(0), the last relation gives
for all x,y∈ R. Replacing x by kx in (2.30) and on solving with the help (2.30) and using the fact that d1(k)=d2(k), we get (xd1(x)−xd2(x))y*∈Z(R) for all x,y ∈ R. Replacing y by h, where h∈H(R)∩Z(R). Then by the primeness of R and S(R)∩Z(R)≠(0) condition force that xd1(x)−xd2(x)∈Z(R) for all x ∈ R. Linearizing this we get xd1(y)+yd1(x)−xd2(y)−yd2(x)∈Z(R) for all x,y ∈ R. Taking y by h where h∈H(R)∩Z(R) and using d1(h)=d2(h), we obtain d1(x)−d2(x)∈Z(R) for all x ∈ R. This can be further written as
[d1(x),r]−[d2(x),r]=0for allx,r∈R.
Replacing x by xr in (2.31), we get [x,r](d1(r)−d2(r))=0 for all x,r ∈ R. Substitute xu for x in the last relation, we obtain [x,r]u(d1(r)−d2(r))=0 for all x,r,u ∈ R. Then by the primeness of R, for each fixed r ∈ R, we get either [x,r]=0 for all x ∈ R or d1(r)−d2(r)=0. Define A={r∈R|[x,r]=0for allx∈R} and B={r∈R|d1(r)−d2(r)=0}. Clearly, A and B are additive subgroups of R whose union is R. Hence by Brauer's trick, either A=R or B=R. If A=R, then [x,r]=0 for all x,r∈ R. This implies that R is commutative. If B=R, then d1(r)=d2(r) for all r∈ R, which is a contradiction to our assumption. Hence, we conclude that R is commutative.
Theorem 2.7. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If R admit derivations d1, d2 such that at least one of them is nonzero and satisfies d1(*[x,x*]2)+*[x,d2(x*)]2∈Z(R) for all x ∈ R, then R is commutative.
Proof. We are given that d1 and d2 are derivations of R such that
d1(*[x,x*]2)+*[x,d2(x*)]2∈Z(R)for allx∈R.
If d2 is zero then by Theorem 2.4, we get R is commutative. If d1 is zero then we have *[x,d2(x*)]2∈Z(R) for all x∈ R. Expansion of last relation gives
x2d2(x*)−2xd2(x*)x*+d2(x*)(x*)2∈Z(R)for allx∈R.
Replacing x by hx, where h∈H(R)∩Z(R) in (2.33) and combining the obtained expression, we get *[x,x*]2d2(h)h2∈Z(R) for all x ∈ R. Now applying the primeness of the ring R, we get either *[x,x*]2∈R or d(h)h2=0. If *[x,x*]2∈Z(R) for all x∈ R, then by Lemma 2.2, we get R is commutative. Now consider the second case in which we have d2(h)h2=0 for all h∈H(R)∩Z(R). This implies that d2(h)=0, from here we get d2(k)=0 for all k∈S(R)∩Z(R). Replacing x by kx in (2.30) and using the fact that d2(k)=0, we get 4xd2(x*)x*k3∈Z(R) for all x∈ R. This implies that xd2(x*)x*∈Z(R) for all x ∈ R. Arguing as above after (2.13), we conclude that R is commutative.
Now consider the second case in which both d1 and d2 are nonzero. On expansion of (2.32), we have
for all x ∈ R. Replacing x by hx, where h∈H(R)∩Z(R) in (2.34) and solving with the help of (2.34), we get
*[x,x*]2(3d1(h)+d2(h))h2∈Z(R)for allx∈R.
By the primeness of the ring R, we get either *[x,x*]2∈Z(R) for all x∈ R or (3d1(h)+d2(h))h2=0 for all h∈H(R)∩Z(R). If *[x,x*]2∈Z(R) for all x ∈ R, then by Lemma 2.2, we get R is commutative. Now consider the case (3d1(h)+d2(h))h2=0. This implies that d2(h)=−3d1(h) and hence d2(k)=−3d1(k) for all k∈S(R)∩Z(R). Now substituting kx for x, where k∈S(R)∩Z(R) in (2.34) and combining the obtained result with (2.34), we get 4(d1(x(x*)2)+xd2(x*)x*)k3∈Z(R) for all x ∈ R. Since R is 2-torsion free ring and S(R)∩Z(R)≠(0), then invoking the primeness of R we obtain d1(x(x*)2)+xd2(x*)x*∈Z(R) for all x∈ Z(R). Linearization to the last expression gives
d1(xx*y*)+d1(xy*x*)+d1(x(y*)2)+d1(y(x*)2)+d1(yx*y*)+d1(yy*x*)+xd2(x*)y*+xd2(y*)x*+xd2(y*)y*+yd2(x*)x*+yd2(x*)y*+yd2(y*)x*∈Z(R) for allx,y∈R.
Subtracting (2.38) form (2.37), we get (−2xx*y*+4xy*x*−2y(x*)2)d1(k)k∈Z(R) for all x,y∈R. Since R is 2-torsion free prime ring and S(R)∩Z(R)≠(0), the last expression forces that either xx*y*−2xy*x*+y(x*)2∈Z(R) for all x,y ∈ R or d1(k)k=0. Suppose
xx*y*−2xy*x*+y(x*)2∈Z(R)for allx,y∈R.
Substituting ky for y, where k∈S(R)∩Z(R) in (2.39) and combining with (2.39), we get 2y(x*)2k∈Z(R) for all x,y ∈ R. Taking x=k, we obtain 2yk3∈Z(R) for all y∈ R. Since R is 2-torsion free prime ring and S(R)∩Z(R)≠(0), we conclude that R is commutative. Now consider the case in which we have d1(k)k=0 for all k∈S(R)∩Z(R). This implies that d1(k)=0 for all k∈S(R)∩Z(R). This further implies that d2(k)=0. Substitute k for x in (2.36), to get
−2d1(y*)k2+d1(y)k2−d2(y*)k2∈Z(R)for ally∈R.
Replacing y by ky, where k∈S(R)∩Z(R) in (2.40) and combining the obtained relation with (2.40), finally we get 2d1(y)k3∈Z(R) for all y∈ R. Since R is 2-torsion free ring and S(R)∩Z(R)≠(0), we obtain d1(y)∈Z(R) for all y ∈ R. Hence, by Posner's [4] first theorem, R is commutative.
As an immediate consequence of the above theorem, we get the following corollary:
Corollary 2.8. Let R be a 2-torsion free prime ring with involution '*' of the second kind. If R admits a nonzero derivation d such that d(*[x,x*]2)+*[x,d(x*)]2∈Z(R) for all x ∈ R, then R is commutative.
The following example shows that the second kind involution assumption is essential in Theorem 2.3 and Theorem 2.4.
Example 2.9. Let R=β1β2β3β4|β1,β2,β3,β4∈ℤ. Of course, R with matrix addition and matrix multiplication is a noncommutative prime ring. Define mappings *,d1,d2:R→R such that β1β2β3β4*=β4−β2−β3β1, d1β1β2β3β4=0−β2β30 and d2β1β2β3β4=0β2−β30
Obviously, Z(R)=β100β1|β1∈ℤ. Then x*=x for all x∈Z(R), and hence Z(R)⊆H(R), which shows that the involution * is of the first kind. Moreover, d1 and d2 are nonzero derivations of R such that *[x,d1(x)]2∈Z(R) and *[x,d1(x)]2−*[x,d2(x)]2∈Z(R) for all x∈ R. However, R is not commutative. Hence, the hypothesis of second kind involution is crucial in Theorem 2.3 & 2.4. Our next example shows that Theorems 2.3 and 2.4 are not true for semiprime rings.
Example 2.10. Let S=R×ℂ, where R is same as in Example 2.9 with involution '*' and derivations d1 and d2 same as in above example, ℂ is the ring of complex numbers with conjugate involution τ. Hence, S is a 2-torsion free noncommutative semiprime ring. Now define an involution α on S, as (x,y)α=(x*,yτ). Clearly, α is an involution of the second kind. Further, we define the mappings D1 and D2 from S to S such that D1(x,y)=(d1(x),0) and D2(x,y)=(d2(x),0) for all (x,y)∈S. It can be easily checked that D1, D2 are derivations on S and satisfying α[X,D1(X)]2∈Z(S) and α[X,D1(X)]2−α[X,D2(X)]2∈Z(S) for all X ∈ S, but S is not commutative. Hence, in Theorems 2.3 & 2.4, the hypothesis of primeness is essential.
We conclude the paper with the following Conjectures.
Conjecture 2.11. Let n>2 be an integer, R be a prime ring with involution '*' of the second kind and with suitable torsion restrictions on R. Next, let d be a nonzero derivation on R such that *[x,d(x)]n∈Z(R) for all x∈ R. Then what we can say about the structure of R or the form of d?
Conjecture 2.12. Let n>2 be an integer, R be a prime ring with involution '*' of the second kind and with suitable torsion restrictions on R. Next, let d be a nonzero derivation on R such that d(*[x,x*]n)∈Z(R) for all x∈ R. Then what we can say about the structure of R or the form of d?
Conjecture 2.13. Let n>2 be an integer, R be a prime ring with involution '*' of the second kind and with suitable torsion restrictions on R. Next, let d be a nonzero derivation on R such that d(*[x,x*]n)+*[x,d(x*)]n∈Z(R) for all x∈ R. Then what we can say about the structure of R or the form of d?
The authors are greatful to the learned referee for carefully reading the manuscript. The valuable suggestions have simplified and clarified the paper greatly.