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 for all x∈ R (for n=1, 2), where 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 , otherwise (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 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 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 , where , and . 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 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 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 for all , 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 for all x∈ R, then R is commutative.
Proof. Linearization of gives that for all Taking x=-x in the last expression and combine it with the above relation, we get
Substituting ky for y, where , we obtain for all Invoking the primeness of R and using the fact that , we get
Combining (2.1) and (2.2), we conclude that Replacing y by y*, we get for all This can be further written as for all x,y,w∈ R. Replacing y by ry, we get for all . 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 for all x ∈ R, then R is commutative.
Substitute ky for y, where in (2.6), we get for all This implies that for all 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 for all , then R is commutative.
proof.Case(i) First we discuss the case, "when n=1" and "i.e.",
Linearizing the above expression, we get
That is,
This further implies that
Hence
Replacing y by hy in (2.7), where and using it, we have
Using the primeness of R, we obtain either d(h)=0 or
First we consider the situation
Substituting kx for x in (2.8), where and combining it with (2.8), we get
Since R is 2-torsion free prime ring, we deduce that
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 . This implies that d(k)=0 for all Replacing y by ky in (2.7), where 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 the above relation implies that
Taking y=h, where and using the fact that d(h)=0, we get [d(x),r]h=0 for all x,r ∈ R and . This yields that [d(x),r]=0 for all x,r ∈ R. Hence in view of Posner's [4] first theorem, R is commutative.
Then by the primeness of R we are force to conclude that either or for all x∈ R. First we consider the case
Substituting kx for x in (2.10), where , we get This further implies that
Subtracting (2.10) from (2.11) and using 2-torsion freeness of R, we obtain Therefore, by Lemma 2.1, R is commutative. Now consider the second case for all . This implies that d(h)=0 for all . Therefore, d(k)=0 for all . Replacing x by kx in (2.9), where and using the fact that d(k)=0, we obtain
Application of (2.9) yields for all x ∈ R. Since R is 2-torsion free ring and , we get
Putting x+h in place of x, where , we arrive at
Taking x=-x in (2.14) and then combining it with the obtained relation, we get . This implies that 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 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.,
Using the primeness of R, we have either d(h)=0 or
for all x,y,r ∈ R. We first consider the relation (2.16). Replacing y by ky, where in (2.16), we get for all . Since R is 2-torsion free ring and , we obtain for all Taking x=k, where , 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 . This implies that d(k)=0 for all Replacing y by ky, where in (2.15) and making use of (2.15), we get
This implies that
Taking x=h where 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 , we obtain [d(y),r]=0 for all y,r∈ R. Hence by Posner's [4] first theorem, R is commutative.
Substituting -x for x in (2.18) and combining the obtained relation with (2.18), we obtain
Since R is 2-torsion free, the last relation gives
Replacing y by hy, where in (2.19) and intermix it with (2.19), we come to
By the primeness of the ring R, we get either d(h)=0 or
for all x,y ∈ R. Replacing y by ky, where in (2.20), we arrive at
This implies that
Substituting kx for x in the last relation, we conclude that 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 " This implies that d(k)=0 for all Now replacing x by kx in (2.19) and using "d(k)=0, we get"
Since , the last expression implies that
Combining this with (2.19) and using the fact that R is 2-torsion free ring, we arrive at
Replacing x by kx, where and combining it with previous expression, we obtain for all x,y ∈ R. Replacing x by h, where we come to for all y ∈ R. This implies that 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 for all x ∈ R, then R is commutative.
Proof. By the hypothesis we assume that
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 , we obtain for all x ∈ R. Now by the primeness of R we get either for all x ∈ R or . Now, we suppose that
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 for all . Since R is prime ring, so we get d(h)=0. This also implies that d(k)=0 for all Replacing x by xk in (2.22) and combining with (2.22) we arrive at for all x∈ R. Since , so by the primeness of R, we get
for all x,y ∈ R.Replacing x by -x in (2.25) and combining the obtained relation with (2.25), we obtain
for all x,y ∈ R. Taking x=h, where , we get
The primeness of R yields that 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 for all x ∈ R, then R is commutative.
Proof. We assume that
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 and on simplifying with the help of (2.27), we get
This implies either for all x ∈ R or . If for all x∈ R, then by using the same steps as we have used after (2.10), we arrive at for all x ∈ R. Thus R is commutative, by Lemma 2.1. On the other hand, if for all Then we are force to conclude that and hence for all Replacing x by kx in (2.27), and combining with (2.27) by using the fact that , we get
Since R is 2-torsion free and the last relation gives
Replacing x by -x in (2.29) and combining the obtained result with (2.29), we get
for all x,y ∈ R. Since R is 2-torsion free ring, the above expression yields
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 , we get for all x,y ∈ R. Replacing y by h, where . Then by the primeness of R and condition force that for all x ∈ R. Linearizing this we get for all x,y ∈ R. Taking y by h where and using we obtain for all x ∈ R. This can be further written as
Replacing x by xr in (2.31), we get for all x,r ∈ R. Substitute xu for x in the last relation, we obtain 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 . Define and . 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 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 for all x ∈ R, then R is commutative.
Proof. We are given that d1 and d2 are derivations of R such that
If d2 is zero then by Theorem 2.4, we get R is commutative. If d1 is zero then we have for all x∈ R. Expansion of last relation gives
Replacing x by hx, where in (2.33) and combining the obtained expression, we get for all x ∈ R. Now applying the primeness of the ring R, we get either or . If for all x∈ R, then by Lemma 2.2, we get R is commutative. Now consider the second case in which we have for all This implies that from here we get for all Replacing x by kx in (2.30) and using the fact that , we get for all x∈ R. This implies that 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 in (2.34) and solving with the help of (2.34), we get
By the primeness of the ring R, we get either for all x∈ R or for all If for all x ∈ R, then by Lemma 2.2, we get R is commutative. Now consider the case . This implies that and hence for all Now substituting kx for x, where in (2.34) and combining the obtained result with (2.34), we get for all x ∈ R. Since R is 2-torsion free ring and , then invoking the primeness of R we obtain for all x∈ Z(R). Linearization to the last expression gives
for all x,y ∈ R. Since R is 2-torsion free ring, we get
for all x,y ∈ R. Substituting ky for y, where in (2.36) and combining with (2.36) with use of , we arrive at
for all x,y ∈ R and . Substitute ky for y in (2.37) yields
Subtracting (2.38) form (2.37), we get for all . Since R is 2-torsion free prime ring and , the last expression forces that either for all x,y ∈ R or . Suppose
Substituting ky for y, where in (2.39) and combining with (2.39), we get for all x,y ∈ R. Taking x=k, we obtain for all y∈ R. Since R is 2-torsion free prime ring and , we conclude that R is commutative. Now consider the case in which we have d1(k)k=0 for all This implies that d1(k)=0 for all . This further implies that . Substitute k for x in (2.36), to get
Replacing y by ky, where in (2.40) and combining the obtained relation with (2.40), finally we get for all y∈ R. Since R is 2-torsion free ring and , we obtain 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 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 Of course, R with matrix addition and matrix multiplication is a noncommutative prime ring. Define mappings such that , and
Obviously, Then for all , and hence , which shows that the involution * is of the first kind. Moreover, d1 and d2 are nonzero derivations of R such that and 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 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 . Clearly, α is an involution of the second kind. Further, we define the mappings D1 and D2 from S to S such that and for all (x,y)∈S. It can be easily checked that D1, D2 are derivations on S and satisfying and 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 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 for all x∈ R. Then what we can say about the structure of R or the form of d?
Conjecture 2.12. Let 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 for all x∈ R. Then what we can say about the structure of R or the form of d?
Conjecture 2.13. Let 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 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.