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### Article

Kyungpook Mathematical Journal 2022; 62(1): 43-55

Published online March 31, 2022

### On n-skew Lie Products on Prime Rings with Involution

Shakir Ali and Muzibur Rahman Mozumder*

epartment of Mathematics, Aligarh Muslim University, Aligarh-202002, U. P., India
e-mail : shakir.ali.mm@amu.ac.in and muzibamu81@gmail.com

Received: July 7, 2021; Revised: October 5, 2021; Accepted: October 7, 2021

### Abstract

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)]nZ(R) for all x∈ R (for n=1, 2), where d:RR 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.

Keywords: Prime ring, derivation, involution, centralizing mappings, 2-skew Lie product, 2-skew centralizing mappings, n-skew commuting mappings, n-skew centralizing mapping

### 1. Introduction

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:RR 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]n1], where *[x,y]0=y, *[x,y]=xyyx* and *[x,y]2=x2y2xyx*+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)]nZ(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)]nZ(R) for all xR (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.

### 2. Main Results

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,yR. Taking x=-x in the last expression and combine it with the above relation, we get

x2y*+xyx*+yxx*Z(R)  for all  x,yR.

Substituting ky for y, where kS(R)Z(R), we obtain (x2y*+xyx*+yxx*)kZ(R) for all x,yR. Invoking the primeness of R and using the fact that S(R)Z(R)(0), we get

x2y*+xyx*+yxx*Z(R)  for all  x,yR.

Combining (2.1) and (2.2), we conclude that x2y*Z(R). Replacing y by y*, we get x2yZ(R) for all x,yR. 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,rR. 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*]2Z(R) for all x ∈ R, then R is commutative.

Proof. By the hypothesis, we have

x2x*2x(x*)2+(x*)3Z(R)  for all  xR.

Replacing x by kx in (2.3) where kS(R)Z(R), we get

x2x*k32x(x*)2k3(x*)3k3Z(R)  for all  xR.

By (2.3) and (2.4), we conclude that 4x(x*)2k3Z(R) for all x∈ R. Since R is 2-torsion free prime ring and S(R)Z(R)(0), we obtain x(x*)2Z(R) for all x∈ R. On linearizing we get

xx*y*+xy*x*+x(y*)2+y(x*)2+yx*y*+yy*x*Z(R)  for all  x,yR.

Taking x=-x in (2.5) and using (2.5), we obtain

xx*y*+xy*x*+y(x*)2Z(R)  for all  x,yR.

Substitute ky for y, where kS(R)Z(R) in (2.6), we get 2y(x*)2kZ(R) for all x,yR. This implies that yx2Z(R) for all x,yR. 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)]nZ(R) for all xR(n=1,2), then R is commutative.

proof. Case (i) First we discuss the case, "when n=1" and "i.e.",

*[x,d(x)]Z(R)  for all  xR.

Linearizing the above expression, we get

*[x,d(y)]+*[y,d(x)]Z(R)  for all  x,yR.

That is,

xd(y)d(y)x*+yd(x)d(x)y*Z(R)  for all  x,yR.

This further implies that

[xd(y),r][d(y)x*,r]+[yd(x),r][d(x)y*,r]=0  for all  x,y,rR.

Hence

x[d(y),r]+[x,r]d(y)d(y)[x*,r][d(y),r]x*+y[d(x),r]+[y,r]d(x)d(x)[y*,r][d(x),r]y*=0  for all  x,y,rR.

Replacing y by hy in (2.7), where hH(R)Z(R) and using it, we have

(x[y,r]+[x,r]yy[x*,r][y,r]x*)d(h)=0  for all  x,y,rR.

Using the primeness of R, we obtain either d(h)=0 or

x[y,r]+[x,r]yy[x*,r][y,r]x*=0  for all  x,y,rR.

First we consider the situation

x[y,r]+[x,r]yy[x*,r][y,r]x*=0  for all  x,y,rR.

Substituting kx for x in (2.8), where kS(R)Z(R) and combining it with (2.8), we get

2(x[y,r]+[x,r]y)k=0  for all  x,y,rR.

Since R is 2-torsion free prime ring, we deduce that

x[y,r]+[x,r]y=0  for all  x,y,rR.

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 hH(R)Z(R). This implies that d(k)=0 for all kS(R)Z(R). Replacing y by ky in (2.7), where kS(R)Z(R) with d(k)=0 and adding with (2.7), we get

2(x[d(y),r]+[x,r]d(y)d(y)[x*,r][d(y),r]x*+y[d(x),r]+[y,r]d(x))k=0

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)=0  for all

x,yR. Taking y=h, where hH(R)Z(R) and using the fact that d(h)=0, we get [d(x),r]h=0 for all x,r ∈ R and hH(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)]2Z(R)  for all  xR.

On expansion we acquire

x2d(x)2xd(x)x*+d(x)(x*)2Z(R)  for all  xR.

Replacing x by xh in (2.9), where hH(R)Z(R), we obtain

(x32x2x*+x(x*)2)d(h)h2Z(R)  for all  xR.

Then by the primeness of R we are force to conclude that either d(h)h2=0 or x32x2x*+x(x*)2Z(R) for all x∈ R. First we consider the case

x32x2x*+x(x*)2Z(R)  for all  xR.

Substituting kx for x in (2.10), where kS(R)Z(R), we get (x3+2x2x*+x(x*)2)k3Z(R)  for all  xR. This further implies that

x3+2x2x*+x(x*)2Z(R)  for all  xR.

Subtracting (2.10) from (2.11) and using 2-torsion freeness of R, we obtain x2x*Z(R)  for all  xR. Therefore, by Lemma 2.1, R is commutative. Now consider the second case d(h)h2=0 for all hH(R)Z(R). This implies that d(h)=0 for all hH(R)Z(R). Therefore, d(k)=0 for all kS(R)Z(R). Replacing x by kx in (2.9), where kS(R)Z(R) and using the fact that d(k)=0, we obtain

x2d(x)k3+2xd(x)x*k3+d(x)(x*)2k3Z(R)  for all  xR.

Application of (2.9) yields 4xd(x)x*k3Z(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 all  xR.

Putting x+h in place of x, where hH(R)Z(R), we arrive at

xd(x)h+d(x)x*h+d(x)h2Z(R)  for all  xR.

Taking x=-x in (2.14) and then combining it with the obtained relation, we get 2d(x)h2Z(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.,

d(*[x,x*])Z(R)  for all  xR.

Linearizing this, we get

d(*[x,y*])+d(*[y,x*])Z(R)  for all  x,yR.

This implies that

d(x)[y*,r]+[d(x),r]y*+x[d(y*),r]+[x,r]d(y*)d(y*)[x*,r][d(y*),r]x*y*[d(x*),r]  [y*,r]d(x*)+d(y)[x*,r]+[d(y),r]x*+y[d(x*),r]+[y,r]d(x*)d(x*)[y*,r]    [d(x*),r]y*x*[d(y*),r][x*,r]d(y*)=0  for all  x,y,rR.

Replacing y by hy, where hH(R)Z(R) in (2.15), we obtain

(x[y*,r]+[x,r]y*y*[x*,r][y*,r]x*+y[x*,r]+[y,r]x*x*[y*,r][x*,r]y*)d(h)=0  for all  x,y,rR.

Using the primeness of R, we have either d(h)=0 or

x[y*,r]+[x,r]y*y*[x*,r][y*,r]x*+y[x*,r]+[y,r]x*x*[y*,r][x*,r]y*=0

for all x,y,r ∈ R. We first consider the relation (2.16). Replacing y by ky, where kS(R)Z(R) in (2.16), we get 2(y[x*,r]+[y,r]x*)k=0 for all x,y,rR. 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,rR. Taking x=k, where kS(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 hH(R)Z(R). This implies that d(k)=0 for all kS(R)Z(R). Replacing y by ky, where kS(R)Z(R) in (2.15) and making use of (2.15), we get

2(d(y)[x*,r]+[d(y),r]x*+y[d(x*),r]+[y,r]d(x*))k=0  for all  x,y,rR.

This implies that

d(y)[x*,r]+[d(y),r]x*+y[d(x*),r]+[y,r]d(x*)=0  for all  x,y,rR.

Taking x=h where hH(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.

Case (ii) Next, for n=2 we have

d(*[x,x*]2)Z(R)  for all   xR.

On expansion we get

d(x2x*)2d(x(x*)2)+d((x*)3)Z(R)  for all  xR.

Linearization of (2.17) yields

d(x2y*)+d(xyx*)+d(xyy*)+d(yxx*)+d(yxy*)+d(y2x*)2d(xx*y*)2d(xy*x*)2d(x(y*)2)2d(y(x*)2)2d(yx*y*)2d(yy*x*)+d((x*)2y*)+d(x*y*x*)+d(x*(y*)2)+d(y*(x*)2)+d(y*x*y*)+d((y*)2x*)Z(R)  for all  x,yR.

Substituting -x for x in (2.18) and combining the obtained relation with (2.18), we obtain

2(d(x2y*)+d(xyx*)+d(yxx*)2d(xx*y*)2d(xy*x*)2d(y(x*)2)+d((x*)2y*)+d(x*y*x*)+d(y*(x*)2))Z(R)  for all  x,yR.

Since R is 2-torsion free, the last relation gives

d(x2y*)+d(xyx*)+d(yxx*)2d(xx*y*)2d(xy*x*)2d(y(x*)2)+d((x*)2y*)+d(x*y*x*)+d(y*(x*)2)Z(R)  for all  x,yR.

Replacing y by hy, where hH(R)Z(R) in (2.19) and intermix it with (2.19), we come to

(x2y*+xyx*+yxx*2xx*y*2xy*x*2y(x*)2+(x*)2y*+x*y*x*+y*(x*)2)d(h)Z(R)  for all  x,yR.

By the primeness of the ring R, we get either d(h)=0 or

x2y*+xyx*+yxx*2xx*y*2xy*x*2y(x*)2+(x*)2y*+x*y*x*+y*(x*)2Z(R)

for all x,y ∈ R. Replacing y by ky, where kS(R)Z(R) in (2.20), we arrive at

2(xyx*+yxx*2y(x*)2)Z(R)  for all  x,yR.

This implies that

xyx*+yxx*2y(x*)2Z(R)  for all  x,yR.

Substituting kx for x in the last relation, we conclude that y(x*)2Z(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 hH(R)Z(R)." This implies that d(k)=0 for all kS(R)Z(R). Now replacing x by kx in (2.19) and using "d(k)=0, we get"

(d(x2y*)+d(xyx*)+d(yxx*)+2d(xx*y*)+2d(xy*x*)2d(y(x*)2)d((x*)2y*)d(x*y*x*)d(y*(x*)2))kZ(R)  for all  x,yR.

Since S(R)Z(R)(0), the last expression implies that

d(x2y*)+d(xyx*)+d(yxx*)+2d(xx*y*)+2d(xy*x*)2d(y(x*)2)d((x*)2y*)d(x*y*x*)d(y*(x*)2)Z(R)  for all  x,yR.

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 all  x,yR.

Replacing x by kx, where kS(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 hH(R)Z(R) we come to d(y)h2Z(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*]2Z(R) for all x ∈ R, then R is commutative.

Proof. By the hypothesis we assume that

*[x,d(x)]2+*[x,x*]2Z(R)  for all  xR.

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

x2d(x)2xd(x)x*+d(x)(x*)2+x2x*2x(x*)2+(x*)3Z(R)  for all  xR.

Replacing x by xh in (2.21), where hH(R)Z(R), we obtain (x3x2x*+x(x*)2)d(h)h2Z(R) for all x ∈ R. Now by the primeness of R we get either x32x2x*+x(x*)2Z(R) for all x ∈ R or d(h)h2=0. Now, we suppose that

x32x2x*+x(x*)2Z(R)  for all  xR.

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 hH(R)Z(R). Since R is prime ring, so we get d(h)=0. This also implies that d(k)=0 for all kS(R)Z(R). Replacing x by xk in (2.22) and combining with (2.22) we arrive at (2xd(x)x*x2x*(x*)3)k3Z(R) for all x∈ R. Since S(R)Z(R)(0), so by the primeness of R, we get

2xd(x)x*x2x*(x*)3Z(R)  for all xR.

Linearization of (2.24) gives

2xd(x)y*+2xd(y)x*+2xd(y)y*+2yd(x)x*+2yd(x)y*+2yd(y)x*x2y*xyx*xyy*yxx*yxy*y2x*(x*)2y*x*y*x*x*(y*)2y*(x*)2y*x*y*(y*)2x*

Z(R) for all x,y ∈ R.Replacing x by -x in (2.25) and combining the obtained relation with (2.25), we obtain

2xd(x)y*+2xd(y)x*+2yd(x)x*x2y*xyx*yxx*(x*)2y*x*y*x*y*(x*)2

Z(R) for all x,y ∈ R. Taking x=h, where hH(R)Z(R), we get

(2d(y)4y*2y)h2Z(R)  for all  yR.

The primeness of R yields that d(y)2y*yZ(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)]2Z(R) for all x ∈ R, then R is commutative.

Proof. We assume that

*[x,d1(x)]2*[x,d2(x)]2Z(R)  for all  xR.

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

x2d1(x)2xd1(x)x*+d1(x)(x*)2x2d2(x)+2xd2(x)x*d2(x)(x*)2Z(R)

for all x ∈ R. Replacing x by xh in (2.27), where hH(R)Z(R) and on simplifying with the help of (2.27), we get

(x32x2x*+x(x*)2)(d1(h)d2(h))h2Z(R)  for all  xR.

This implies either x32x2x*+x(x*)2Z(R) for all x ∈ R or (d1(h)d2(h))h2=0. If x32x2x*+x(x*)2Z(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 hH(R)Z(R). Then we are force to conclude that d1(h)=d2(h) and hence d1(k)=d2(k) for all kS(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*)k3Z(R)  for all  xR.

Since R is 2-torsion free and S(R)Z(R)(0), the last relation gives

xd1(x)x*xd2(x)x*Z(R)  for all  xR.

Linearizing (2.28), we obtain

xd1(x)y*+yd1(x)x*+yd1(x)y*+xd1(y)x*+xd1(y)y*+yd1(y)x*xd2(x)y*xd2(y)x*  xd2(y)y*yd2(x)x*yd2(x)y*yd2(y)x*Z(R)  for all  x,yR.

Replacing x by -x in (2.29) and combining the obtained result with (2.29), we get

2(xd1(x)y*+yd1(x)x*+xd1(y)x*xd2(x)y*xd2(y)x*yd2(x)x*)Z(R)

for all x,y ∈ R. Since R is 2-torsion free ring, the above expression yields

xd1(x)y*+yd1(x)x*+xd1(y)x*xd2(x)y*xd2(y)x*yd2(x)x*Z(R)

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 hH(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 hH(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]=0  for all  x,rR.

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={rR | [x,r]=0 for all xR} and B={rR | 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*)]2Z(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*)]2Z(R)  for all  xR.

If d2 is zero then by Theorem 2.4, we get R is commutative. If d1 is zero then we have *[x,d2(x*)]2Z(R) for all x∈ R. Expansion of last relation gives

x2d2(x*)2xd2(x*)x*+d2(x*)(x*)2Z(R)  for all  xR.

Replacing x by hx, where hH(R)Z(R) in (2.33) and combining the obtained expression, we get *[x,x*]2d2(h)h2Z(R) for all x ∈ R. Now applying the primeness of the ring R, we get either *[x,x*]2R or d(h)h2=0. If *[x,x*]2Z(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 hH(R)Z(R). This implies that d2(h)=0, from here we get d2(k)=0 for all kS(R)Z(R). Replacing x by kx in (2.30) and using the fact that d2(k)=0, we get 4xd2(x*)x*k3Z(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

d1(x2x*)2d1(x(x*)2)+d1((x*)3)+x2d2(x*)2xd2(x*)x*+d2(x*)(x*)2Z(R)

for all x ∈ R. Replacing x by hx, where hH(R)Z(R) in (2.34) and solving with the help of (2.34), we get

*[x,x*]2(3d1(h)+d2(h))h2Z(R)   for all  xR.

By the primeness of the ring R, we get either *[x,x*]2Z(R) for all x∈ R or (3d1(h)+d2(h))h2=0 for all hH(R)Z(R). If *[x,x*]2Z(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 kS(R)Z(R). Now substituting kx for x, where kS(R)Z(R) in (2.34) and combining the obtained result with (2.34), we get 4(d1(x(x*)2)+xd2(x*)x*)k3Z(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 all  x,yR.

Replacing x by -x in (2.35), we get

2(d1(xx*y*)+d1(xy*x*)+d1(y(x*)2)+xd2(x*)y*+xd2(y*)x*+yd2(x*)x*)Z(R)

for all x,y ∈ R. Since R is 2-torsion free ring, we get

d1(xx*y*)+d1(xy*x*)+d1(y(x*)2)+xd2(x*)y*+xd2(y*)x*+yd2(x*)x*Z(R)

for all x,y ∈ R. Substituting ky for y, where kS(R)Z(R) in (2.36) and combining with (2.36) with use of d2(k)=3d1(k), we arrive at

2d1(y(x*)2)k+2yd2(x*)x*kxx*y*d1(k)+y(x*)2d1(k)+2xy*x*d1(k)Z(R)

for all x,y ∈ R and kS(R)Z(R). Substitute ky for y in (2.37) yields

2d1(y(x*)2)k2+2yd2(x*)x*k2+xx*y*d1(k)k+y(x*)2d1(k)k2xy*x*d1(k)k+2y(x*)2d1(k)kZ(R)  for all  x,yR.

Subtracting (2.38) form (2.37), we get (2xx*y*+4xy*x*2y(x*)2)d1(k)kZ(R) for all x,yR. 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*)2Z(R) for all x,y ∈ R or d1(k)k=0. Suppose

xx*y*2xy*x*+y(x*)2Z(R)  for all  x,yR.

Substituting ky for y, where kS(R)Z(R) in (2.39) and combining with (2.39), we get 2y(x*)2kZ(R) for all x,y ∈ R. Taking x=k, we obtain 2yk3Z(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 kS(R)Z(R). This implies that d1(k)=0 for all kS(R)Z(R). This further implies that d2(k)=0. Substitute k for x in (2.36), to get

2d1(y*)k2+d1(y)k2d2(y*)k2Z(R)  for all  yR.

Replacing y by ky, where kS(R)Z(R) in (2.40) and combining the obtained relation with (2.40), finally we get 2d1(y)k3Z(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*)]2Z(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. Of course, R with matrix addition and matrix multiplication is a noncommutative prime ring. Define mappings *,d1,d2:RR such that β1β2β3β4*=β4β2β3β1, d1β1β2β3β4=0β2β30 and d2β1β2β3β4=0β2β30

Obviously, Z(R)=β100β1| β1. Then x*=x for all xZ(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)]2Z(R) and *[x,d1(x)]2*[x,d2(x)]2Z(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×, where R is same as in Example 2.9 with involution '*' and derivations d1 and d2 same as in above example, 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)]2Z(S) and α[X,D1(X)]2α[X,D2(X)]2Z(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)]nZ(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*)]nZ(R) for all x∈ R. Then what we can say about the structure of R or the form of d?

### Acknowledgments

The authors are greatful to the learned referee for carefully reading the manuscript. The valuable suggestions have simplified and clarified the paper greatly.

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