On certain functional equation related to derivations

: In this article, we prove the following result. Let ≥ n 3 be some ﬁ xed integer and let R be a prime ring with ≠ + − R n char

Abstract: In this article, we prove the following result.Let ≥ n 3 be some fixed integer and let R be a prime ring with ) .Suppose there exists an additive mapping 1 Introduction Let R be an associative ring.Given an integer > n 1, a ring R is said to be n-torsion free if, for ∈ x R, the condition = nx 0 implies = x 0. As usual, the commutator − xy yx will be denoted by x y , [ ].A ring R is said to be prime if, for ∈ a b R , ,the condition = aRb 0 implies that either = a 0 or = b 0, and is said to be semiprime in case = aRa 0 ( ) implies = a 0. We denote by R char( ), the characteristic of a prime ring R.An ( ) is fulfilled for all ∈ x R. Clearly, every derivation is a Jordan derivation.The converse is in general not true.A classical result of Herstein [1] asserts that any Jordan derivation on a prime ring with ≠ R char 2 ( ) is a derivation (see [2] for a brief proof).Cusack [3] generalized Herstein theorem to 2-torsion free semiprime rings (see [4] for an alternative proof, and [5] for a proof in the setting of algebras).Beidar et al. [6] have fairly generalized Herstein theorem.For results related to Herstein theorem, the reader is referred to [7][8][9].Brešar [10] has proved the following result (see [11] for a generalization).
Theorem 1.Let R be a 2-torsion free semiprime ring and let → D R R : be an additive mapping satisfying the relation for all pairs ∈ x y R , .In this case, D is a derivation.
An additive mapping D, which maps an arbitrary ring R into itself and satisfies relation (1) for all pairs ∈ x y R , , is called a Jordan triple derivation.One can easily prove that any Jordan derivation on an arbitrary 2torsion free ring is a Jordan triple derivation, which means that Theorem 1 generalizes Cusack's generalization of Herstein theorem.
The above result represents a motivation for many other results (see [12][13][14]).Vukman [15] conjectured that in case there exists an additive mapping → D R R : , where R is a 2-torsion free semiprime ring, satisfying the relation for all pairs ∈ x y R , , then D is a derivation.By our knowledge this conjecture is in general still an open problem.Putting x for y in relations ( 1) and ( 2) we obtain and Recently, Fošner et al. [8] proved the following result regarding relation (4), which is related to Vukman's conjecture mentioned above.
Theorem 2. Let R be a prime ring with ≠ R char 2 ( ) and let → D R R : be an additive mapping satisfying the relation for all ∈ x R. In this case, D is a derivation.
Any Jordan derivation can be written in the form + = + + + D xy yx D x y xD y D y x yD x , ( ) ( ) ( ) ( ) ( ) which gives, putting = y x 2 , relation (4).Therefore, one can conclude that Theorem 2 generalizes Herstein theorem.Relation (3) leads to the following functional equation: which was studied on prime rings by Beidar et al. [6].
2 for y in (2) we obtain after some calculations (see [16] for the details) the following functional equation: where ≥ n 3 is some fixed integer.It is our aim in this article to prove the following result.
Theorem 3. Let ≥ n 3 be some fixed integer and let R be a prime ring with ) , and let → D R R : be an additive mapping satisfying the relation for all ∈ x R. In this case, D is a derivation.
In case = n 3, Theorem 3 reduces to Theorem 2. For functional equations related to derivations we refer to [17][18][19][20] where further references can be found.

Main results
In the proof of Theorem 3, we shall use as the main tool the theory of functional identities (Beidar-Brešar-Chebotar theory).The theory of functional identities considers set-theoretic mappings on rings that satisfy some identical relations.When treating such relations one usually concludes that the form of the maps involved can be described, unless the ring is very special.We refer the reader to [21] for an introductory account on functional identities and to [22] for full treatment of this theory.
We would also like to mention the latest relevant article by Brešar introducing the theory of functional identities and their applications [23].
Let R be a ring and let X be a subset of R. By C X ( ) we denote the set ∈ be arbitrary mappings.In the case when = m 1, this should be understood as that E is an element in R and = p 0.
be arbitrary mappings.Consider functional identities A natural possibility when (7) and ( 8) are fulfilled is when there exist mappings for all ∈ x X m m , ∈ i I , ∈ j J .We shall say that every solution of the form ( 9) is a standard solution of (7) and (8).The case when one of the sets I or J is empty is not excluded.The sum over the empty set of indexes should be simply read as zero.So, when = J 0 (resp.= I 0), (7) and ( 8) reduce to On certain functional equation related to derivations  3 In that case the (only) standard solution is A d-freeness of X will play an important role in this article.For a definition of d-freeness we refer the reader to [24].Under some natural assumptions one can establish that various subsets (such as ideals, Lie ideals, the sets of symmetric or skew symmetric elements in a ring with involution) of certain types of rings are d-free.We refer the reader to [25] and [26] for results of this kind.Let us mention that a prime ring R is a d-free subset of its maximal right ring of quotients, unless R satisfies the standard polynomial identity of degree less than d 2 (see [26,Theorem 2.4]).
Let R be an algebra over a commutative ring ϕ and let be a fixed multilinear polynomial in noncommuting indeterminates x i over ϕ.Here, S n stands for the sym- metric group of order n.Let be a subset of R closed under p, i.e., ∈ p x ¯n ( ) , where

(
) .We shall consider a mapping .Let us mention that the idea of considering the expression p x p y , n n [ ( ) ( )] in its proof is taken from [27].For the proof of Theorem 3, we need Theorem 4, which might be of independent interest.
and therefore where In particular, for all ∈ x y ¯, n n n .
Let us introduce the mapping for the ease of writing and presenting the results.
It is easy to verify that for 1, , we have (by ( 16)) On certain functional equation related to derivations  5 for all ∈ x y ¯, n n n .In exactly the same way for = . Therefore, we have , for all ∈ x y ¯, n n n .
Equation ( 16) can now be rewritten as for all ∈ x y ¯, n n n .
On other way, the last equation can now be rewritten as for all ∈ x y ¯, n n n .Comparing relations (19) and ( 20) we arrive at Let → s : be a mapping defined by = − s i i n.

( )
For each ∈ σ S n the mapping } will be denoted by σ ¯.In the continuation of the article we will write for = i n 1, 2,…, .Now using the theory of functional identities and exposing x π n ( ) in ( 21) from the right side we obtain …, ,  , …, , and ( ).Again using the theory of functional identities a few more times we can conclude after − n 3 ¯π for all ( ).Last equation can now be rewritten as On certain functional equation related to derivations  9 ) ( ).Using the theory of functional identities and first exposing x π 2 ( ) from the right side and then x σ n ¯2 ( ) in ( 22) we obtain Again using the theory of functional identities and exposing ) we obtain where → t u v R , , : 2  and → κ C : 3  ( ).
in the above relation gives where → t u v R , , : 2  , and → κ C : 3  ( ).After the complete linearization of the above identity and considering that is a 6-free subset of R, we obtain where and The next equation is clearly proved in the same way as the last one.Using the theory of functional identities and first exposing from the left side we obtain where , and ′ → λ C : 2  ( ).Replacing the roles of denotations x and y in ( 24) and comparing the so-obtained identities leads to = ) for all ∈ x y , .Putting x for y in ( 24) leads to Using the same arguments, it follows from (25 ) for all ∈ x y , .Therefore, Comparing the above relations gives Hence, there exist ∈ r R and → μ C : ( ) such that Now setting x n instead of y in (24) and multiplying this equation with − 2 n 3 we obtain and so On the other hand, setting + n 1 instead of n into (6) and multiplying this equation with δ n we obtain Now comparing equations ( 28) and ( 29) we obtain On certain functional equation related to derivations  11 Multiplying last equation with 4 and considering .
Using equation (32) into equation (31) we obtain and so The complete linearization of this relation and using the theory of functional identities leads to = Again, after the complete linearization of the last relation and using the theory of functional identities leads to Whence it follows that = − y r , 0

( ) ( ) ( )
And so that D is a Jordan derivation.By Herstein theorem, D is a derivation, which completes the proof of the theorem.□ We are now in the position to prove Theorem 3.
Proof of Theorem 3. The complete linearization of (6) gives us (14).First suppose that R is not a PI ring (satisfying the standard polynomial identity of degree less than 6).According to Theorem 4 the mapping D is a derivation.Assume now that R is a PI ring.It is well known that in this case R has a nonzero center (see [1]).Let c be a nonzero central element.
Picking any ∈ x R and setting = = x x x

Theorem
Complete linearization of the last equation and using the theory of functional identities (exposing everything from the right side) we obtain

Now setting c 3
instead of x in (54) we obtain .