A Note on Generalized Jordan Derivations in Semiprime Rings

The main purpose of this paper is to study and investigate some results concerning generalized Jordan derivation and generalized derivation G:R→R on semiprime ring R, where D an additive mapping on R such that D(xxnn)=∑ xxnn−jjD(xxnn) jj=1 xxjj−1 for all x ∈ R and D acts as left centralizer.


Introduction
The notion of generalized Jordan derivations on rings was introduced by Nakajimain [15].It is a unified and generalized description of generalized Jordan derivations and generalized derivations. A classical result of Herstein [6] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of Herstein's result can be found in [1].Cusack [5] generalized Herstein's result to 2-torsion free semiprime rings (see also [2] for an alternative proof). An additive mapping D:R→ R is called Jordan triple derivation in case D(xyx)=D(x)yx+xD(y)x+xyD(x) holds for all pairs x, y∈ R. Bresar [3] has proved that any Jordan triple derivation on 2-torsion free semiprime ring is a derivation. One can easily prove that any Jordan derivation of arbitrary ring is Jordan triple derivation (see for example [1] for the details) which means that the result we have just mentioned generalized Cusack's generalization of Herstein's theorem. An additive mapping T: R → R is called a left centralizer in case T(xy) = T(x)y holds for all pairs x, y∈ R. An additive mapping T: R→ R is called left Jordan centralizer in case T(x 2 ) = T(x)x holds for all x ∈ R. The definition of a right centralizer and a right Jordan centralizer should be self-explanatory. Obviously, any left centralizer is a left Jordan centralizer. Molnar [8] has proved the following result. Let R be a 2-torsion free prime ring and let T:R → R be an additive mapping. If T(xyx) = T(x)yx holds for every x, y ∈ R, then T is a left centralizer. The concept of generalized derivation has been introduced by Bresar [4]. It is easy to see that F : R → R is a generalized derivation iff F is of the form F = D+ T, where D is a derivation and T a left centralizer. Jing and Lu [7] introduced a concept of generalized Jordan derivation and generalized Jordan triple derivation. An additive mapping F:R→ R is generalized Jordan derivation if F(x 2 )= F(x)x+ xD(x) holds for all x ∈ R where D:R→ R is a Jordan derivation. Kun-Shan Liu [13]proved , let R be a prime ring, let I be a nonzero ideal of R and let n be a fixed positive integer. If the characteristic of R is either 0 or a prime p that is greater than 2n, then an additive map d that satisfies d(x n+1 )=∑ n j=0 x n−j d(x)x j for all x∈ I must be a derivation. Recently N. M. Ghosseiri [14] proved , let R be a 2-torsion free ring with identity and let n ≥ 2.Then(i) any Jordan left derivation (hence, any left derivation) D on the ring M n (R) is identically zero;(ii) any generalized left derivation on M n (R) is a right centralizer, the full matrix ring M n (R) (n ≥2) is identically zero .The main purpose of this paper is to study and investigate some results concerning generalized Jordan derivation and generalized derivation G:R→R on semiprime ring R.

Preliminaries
Throughout, R will represent an associative ring. Given an integer n > 1, a ring R is said to be n-torsion-free if for x ∈ R, nx= 0 implies that x = 0. Recall that a ring R is prime if for a,b ∈ R, aRb= (0) implies that either a= 0 or b= 0, and is semiprime in case aRa= (0) implies that a= 0.Every prime is semiprime but the converse is not true always. An additive mapping D: R→R is called a derivation if D(xy)=D(x)y+ xD(y) holds for all pairs x,y ∈ R and is called a Jordan derivation in case D(x 2 )=D(x)x+ xD(x) is fulfilled for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. An additive mapping F:R→ R is generalized Jordan derivation if F(x 2 )= F(x)x+ xD(x) holds for all x ∈ R where D:R → R is a Jordan derivation, and is called generalized derivation if F(xy)= F(x)y+xD(y) holds for all x,y ∈ R, where D:R→R is a derivation.
The following Lemma are necessary for the paper. Let R be a semiprime ring of characteristic not two and T:R → R an additive mapping which satisfies T(x 2 ) = T(x)x for all x ∈ R. Then T is a left centralizer.

Theorem 3.1
Let n > 1 be an integer and let R be a n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mappings D,G:R→R such that for all x∈ R and D acts as left centralizer if G(x 2 )=G(x)x+D( +1 ) for all x∈ R, then G is Jordan generalized derivation on R.
Proof: From the relation G(x 2 )= G(x)x + D( n+1 ) for all x∈ R,with using that D,G acts as right and left centralizer respectively, we obtain G(x 2 )=G(x)x+xD( n ) for all x∈ R. For complete our proof, we must prove that D is derivation. According our hypothesis, we assume that D for all x∈ R. It follows immediately that D(e)= 0, where e denotes the identity element. Putting x + c for x in the above relation, where c is any element of the center Z(R) such that D(c)= 0,we obtain )for all x∈ R.
We adopt the convention that x 0 = e for all x ∈ R.
for all x ∈ R ,by using this relation and rearranging (1) in the sense of collecting together terms involving equal number of factors of c, we obtain for all x ∈ R.
where ᵢ( , ) stands for the expression of terms involving i factors of c. We replace c by e,2e, 3e, . . ., (n−1)e in turn in (2). Expressing the resulting system of n−1 homogeneous equations, we see that the coefficient matrix of the system is a van der Monde matrix Since the determinant of the matrix is different from zero, it follows that the system has only a trivial solution. In particular, Then the above equation reduces to Thus , from above relation we have Since R is n!-torsion-free, it follows from the relation (3) In other words, D is a Jordan derivation. As we have already mentioned, any Jordan derivation on a 2-torsion-free semiprime ring is a derivation. Now, we can apply the fact in the relation G(x 2 )=G(x)x+xD( ) for all x∈ R. The proof of the theorem is complete.

Theorem 3.2
Let n > 1 be an integer and let R be a n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mappings D,G:R→R such that for all x∈ R and D acts as left centralizer if G(xy)=G(x)y+D( ) for all x, ∈ R, then G is generalized derivation on R.
Proof: From the relation G(xy)=G(x)y+D( ) for all x, ∈ R,with using that D,G acts as right and left centralizer respectively, we obtain G(xy)=G(x)y+xD( n ) for all x, ∈ R.By same process in Theorem3.1,we complete our proof.

Corollary 3.3
Let n > 1 be an integer and let R be a n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mappings D,G:R→R such that D( )=∑ − D( ) (ii)G(xy)=G(x)y+D( ) for all x, ∈ R.Then D is Jordan derivation( resp. derivation )on R.

Theorem 3.4
Let n > 1 be an integer and let R be a n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mappings D,G:R→R such that D( )=∑ − D( ) =1 −1 for all x∈ R and D acts as left centralizer if G(x 2 )=G(x)x+D( +1 ) for all x∈ R,then G is generalized derivation on R.
Proof: We have from Theorem3.1, that G is generalized Jordan derivation, therefore, the relation G(x 2 )=G(x)x+xD(x) for all x∈ R,where G is a Jordan derivation of R. Since R is a semiprime ring one can conclude that D is a derivation. Let us denote G-D by T.Then we have T(x 2 ) = G(x 2 )−D(x 2 ) = G(x)x+xD(x)-D(x)x-xD(x)=(G(x)-D(x))x=T(x)x.We have therefore T(x 2 )=T(x)x, for all x ∈ R. In other words, T is a left Jordan centralizer of R. Since R is a 2-torsion free semiprime ring one can conclude that T is a left centralizer by Lemma A. Hence G is of the form G = D+T, where D is a derivation and T is a left centralizer of R, which means that G is a generalized derivation. The proof is complete.