On certain functional equation related to derivations

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 n x = 0 implies x = 0 . As usual, the commutator x y − y x will be denoted by [ x , y ] . A ring R is said to be prime if, for a , b ∈ R , the condition a R b = 0 implies that either a = 0 or b = 0 , and is said to be semiprime in case a R a = ( 0 ) implies a = 0 . We denote by char ( R ) , the characteristic of a prime ring R . An additive mapping D : R → R is called a derivation if the equation D ( x y ) = D ( x ) y + x D ( y ) holds for all x , y ∈ R . A derivation D is inner in case there exists a ∈ R such that D ( x ) = [ a , x ] holds for all x ∈ R . An additive mapping D : R → R is called a Jordan derivation in case D ( x 2 ) = D ( x ) x + x D ( x ) 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 char ( R ) ≠ 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 [7–9]. Brešar [10] has proved the following result (see [11] for a generalization).

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 2-torsion 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 [12–14]). 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.

Any Jordan derivation can be written in the form D ( x y + y x ) = D ( x ) y + x D ( y ) + D ( y ) x + y D ( 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].

Putting x 2 , x 3 , … , x n − 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.

In case n = 3 , Theorem 3 reduces to Theorem 2. For functional equations related to derivations we refer to [17–20] where further references can be found.

2 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 { r ∈ R ∣ [ r , X ] = 0 } . Let m ∈ N and let E : X m − 1 → R , p : X m − 2 → R be arbitrary mappings. In the case when m = 1 , this should be understood as that E is an element in R and p = 0 . Let 1 ≤ i < j ≤ m and define E i , p i j , p j i : X m → R by

where x ¯ m = ( x 1 , … , x m ) ∈ X m .

Let I , J ⊆ { 1 , … , m } , and for each i ∈ I , j ∈ J let E i , F j : X m − 1 → R be arbitrary mappings. Consider functional identities

A natural possibility when (7) and (8) are fulfilled is when there exist mappings p i j : X m − 2 → R , i ∈ I , j ∈ J , i ≠ j , and λ k : X m − 1 → C ( X ) , k ∈ I ∪ J , such that

for all x ¯ m ∈ X 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

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 2 d (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 symmetric group of order n . Let ℒ be a subset of R closed under p , i.e., p ( x ¯ n ) ∈ ℒ for all x 1 , x 2 , … , x n ∈ ℒ , where x ¯ n = ( x 1 , x 2 , … , x n ) . We shall consider a mapping D : ℒ → R satisfying

for all x 1 , x 2 , … , x n ∈ ℒ . Let us mention that the idea of considering the expression [ p ( x ¯ n ) , p ( y ¯ n ) ] in its proof is taken from [27]. For the proof of Theorem 3, we need Theorem 4, which might be of independent interest.