CLASSIFICATION OF SOME SPECIAL GENERALIZED DERIVATIONS

The purpose of the present paper is to classify generalized derivations satisfying more specific algebraic identities in a prime ring with involution of the second kind. Some well-known results characterizing commutativity of prime rings by derivations have been generalized by using generalized derivation. Mathematics Subject Classification (2020): 16N60, 16W10, 16W25

Lanski generalizes the result of Posner by considering a derivation d such that [d(x), x] ∈ Z(R) for all x in a nonzero Lie ideal U of R.
More recently several authors consider similar situation in the case the derivation d is replaced by a generalized derivation. More specifically an additive map F : R −→ R is said to be a generalized derivation if there exists a derivation d of R such that F (xy) = F (x)y + xd(y) for all x, y ∈ R. Basic examples of generalized derivations are the usual derivations on R and left R-module mappings from R into itself. An important example is a map of the form F (x) = ax + xb for some elements a, b ∈ R; such generalized derivations are called inner. Generalized derivations have been primarily studied on operator algebras thus any investigation from the algebraic point of view might be interesting (see for example [6], [7] and [11]).
During the last two decades, many authors have studied commutativity of prime and semi-prime rings admitting suitably constrained additive mappings acting on appropriate subsets of the rings. Moreover, many of obtained results extend other ones previously proven just for the action of the considered mapping on the whole ring. In this direction, the recent literature contains numerous results on commutativity in prime and semi-prime rings admitting suitably constrained derivations and generalized derivations, and several authors have improved these results by considering rings with involution (for example, see [1], [2], [13] for all x, y, z ∈ R, and F = 0, then there exists λ in the extended centroid of R such that F (x) = λG(x) for all x ∈ R. Lemma 2.3. Let R be a 2-torsion free prime ring and F : R → R be a generalized derivation associated with a derivation d. Then the following assertions are equivalent: (4) There exists λ in the extended centroid of R such that F (x) = λx for all x ∈ R (and therefore d = 0).
Proof. It is enough to prove that each of (1), (2) and (3) implies (4). We first recall that the generalized derivation F is of the form F (x) = λx + d(x) for all x ∈ R and some λ in the maximal left ring of quotients Q(R) of R by [11].
In [3,Theorem 2.9] it is proved that if R is a 2-torsion free semi-prime ring with a generalized derivation F associated with a nonzero derivation d such that R contains a nonzero central ideal.
Motivated by [3], our purpose is to study the same identity on prime rings with involution. More precisely the following theorem classifies generalized derivations satisfying such condition.
Theorem 2.4. Let (R, * ) be a 2-torsion free prime ring with involution of the second kind. If R admits a generalized derivation F associated with a derivation d, then the following assertions are equivalent: (3) There exists λ in the extended centroid of R such that F (x) = λx for all Proof. We need only prove that (1) ⇒ (3) and (2) ⇒ (3).
(1) ⇒ (3) We are given that Linearizing the above relation we find that

M. A. IDRISSI AND L. OUKHTITE
Replacing y by yh in (2), where h ∈ Z(R) ∩ H(R)\{0}, one can obtain Since R is prime, then Eq. Taking From EQs. (2) and (4) it follows that By view of Lemma 2.3, there exists λ in the extended centroid of R such that Replacing x by x + y, we obtain thereby obtaining Replacing y by yh in (7), where h ∈ Z(R) ∩ H(R)\{0}, it is obvious to see that In light of primeness, it follows that Substituting ys for y in (7), with 0 = s ∈ Z(R) ∩ S(R), one can obtain Comparing (7) with (8), we find that Another use of Lemma 2.3, gives the required result.
In [3,Theorem 2.8] it is proved that if R is a 2-torsion free semi-prime ring with a generalized derivation F associated with a nonzero derivation d such that where I is a nonzero ideal of R, then R contains a nonzero central ideal.
Our aim in the following theorem is to study the case the identity ( ) is replaced by a more general algebraic identity. More precisely, we classify the generalized derivation.
Theorem 2.5. Let (R, * ) be a 2-torsion free prime ring with involution of the second kind and F a generalized derivation associated with a derivation d such that Proof. Assume that By linearization we get which leads to Taking y = yh in (11), where h ∈ Z(R) ∩ H(R)\{0}, we obtain [x, y]d(h) = 0 for all x, y ∈ R.
Since R is prime, it follows that either d(h) = 0 or R is commutative.
Assume that d(h) = 0 for all h ∈ Z(R) ∩ H(R); writing ys instead of y in (11) with Using (11) together with (12), we find that In view of Lemma 2.3, there exists λ in the extended centroid of R such that In [10, Theorems 2.3 and 2.4] it is proved that if R is a 2-torsion free semiprime ring admitting a generalized derivation F with associated nonzero derivation d satisfying any one of the following conditions for all x, y in a nonzero ideal I of R, then R contains a nonzero central ideal.
Our next purpose in the following theorem is to study generalized derivations (F, d) satisfying the above identities in the case of prime rings with involution. We have studied this problem and proved that such conditions cannot be considered as commutativity criteria. Moreover, we successfully provide a complete description of those generalized derivations.
Theorem 2.6. Let (R, * ) be a 2-torsion free prime ring with involution of the second kind. If R admits a generalized derivation F associated with a derivation d, then the following assertions are equivalent: (3) F = 0.
Proof. (1) ⇒ (3) By the assumption, we have A linearization of (14) yields and thus Replacing y by yh in (15), where h ∈ Z(R) ∩ H(R)\{0}, one can easily verify that In view of primeness, the above expression yields that either xy − x * y * = 0 or d(h) = 0.

Suppose that
xy − x * y * = 0 for all x, y ∈ R.
(2) ⇒ (3) We are assuming that Linearizing the above relation, one can see that and therefore Replacing y by yh in (21), where h ∈ Z(R) ∩ H(R)\{0}, we obtain then xy + x * y * d(h) = 0 for all x, y ∈ R.
Arguing as above, equation (23) implies that Replace y by ys in (21), where s ∈ Z(R) ∩ S(R)\{0}, we obtain Adding relations (21) and (24), we get Replacing y by h in (25), we find that Then we conclude that F = 0.
As an application of our result, the following theorem constitute a suitable ver- for all x, y ∈ R; (3) F = 0.
The following theorem provides some commutativity criteria for prime rings with involution involving generalized derivations. Furthermore, we classify such generalized derivations.
Theorem 2.8. Let (R, * ) be a 2-torsion free prime ring with involution of the second kind. If R admits a nonzero generalized derivation F associated with a derivation d satisfying one of the following conditions : then R is commutative. Furthermore, there exists λ in the extended centroid of R such that F (x) = λx for all x ∈ R.
By view of Lemma 2.2, there exists λ in the extended centroid of R such that (2) Suppose that A linearization of (34) leads to Replacing y by ys in (35), where 0 = s ∈ Z(R) ∩ S(R), we arrive at Comparing (35) with (38), it follows that Writing yx instead of y in (39) and invoking (39), we obtain xyd(x) = 0 so that d = 0. Therefore our identity reduces to F [x, x * ] = 0 for all x ∈ R. Using the same techniques as used above we conclude that R is commutative and d = 0. Finally, there exists λ in the extended centroid of R such that F (x) = λx for all x ∈ R.

M. A. IDRISSI AND L. OUKHTITE
A linearization of (40) implies that Replacing y by yh in (41), where h ∈ Z(R) ∩ H(R)\{0}, it is obvious to verify that Since equation (42) is the same as equation (16), arguing as above, we are forced to conclude d(h) = 0.
Replacing y by ys in (41), where s is a nonzero element in Z(R) ∩ S(R), we have Combining (41) with (43), one has Writing h instead of y in (44), it follows that which proves that d = 0, and (40) becomes F ([x, x * ]) = 0 for all x ∈ R. Consequently, d = 0, R is commutative and there exists λ in the extended centroid of R such that F (x) = λx for all x ∈ R.
As an application of Theorem 2.8, we have the following result. then R is commutative. Furthermore, there exists λ in the extended centroid of R such that F (x) = λx for all x ∈ R.
The following example proves that the primeness hypothesis in Theorem 2.8 is not superfluous. Set R = R × C, then it is obvious to verify that (R, σ) is a semi-prime ring with involution of the second kind where σ(r, z) = (r * ,z).
Moreover, if we put F(r, z) = (F (r), 0) then F is a left multiplier satisfying the conditions of Theorems 2.8 but R is not commutative.
The following example proves that the condition * is of the second kind is necessary in Theorem 2.8. is a left multiplier that satisfies conditions of Theorem 2.8 however R is not commutative.